This book describes the exemplary CMA Activities delivered with Coach 6.23 Studio MV. November 2007, version 1.0 Hardware and software are distributed by the CMA foundation. The CMA foundation is affiliated to the AMSTEL Institute of Universiteit van Amsterdam. AMSTEL Institute/CMA Foundation Kruislaan 404, 1098 SM Amsterdam, The Netherlands Telephone: +31 20 5255869 Fax: +31 20 5255866 E-mail: [email protected] Internet: http://www.cma.science.uva.nl/english © CMA / AMSTEL Institute, Amsterdam, Text: Ewa KĊdzierska, Vincent Dorenbos, ¤2007 Foundation CMA/AMSTEL Institute, Universiteit van Amsterdam 2 Contents at a glance PREFACE ................................................................................................................................... 4 I. DATA VIDEO ACTIVITIES ......................................................................................................... 5 1. INTRODUCTION...................................................................................................................................................... 6 2. INTRODUCTORY DATA VIDEO ACTIVITIES ............................................................................................................ 7 3. DATA VIDEO EXAMPLES ..................................................................................................................................... 20 4. INTRODUCTORY DATA IMAGE ACTIVITIES .......................................................................................................... 26 5. DATA IMAGE EXAMPLES ..................................................................................................................................... 30 II. MODELING ACTIVITIES ........................................................................................................ 35 1. INTRODUCTION.................................................................................................................................................... 36 2. INTRODUCTORY MODELING ACTIVITIES ............................................................................................................. 37 3. BIOLOGY MODELS ............................................................................................................................................... 44 4. CHEMISTRY MODELS ........................................................................................................................................... 53 5. PHYSICS MODELS................................................................................................................................................. 58 3 Preface Coach 6 Studio MV is a versatile Learning and Authoring Environment for Science, Mathematics and Technology which integrates tools needed for: - capturing videos - measuring on images and video clips - modeling dynamic systems - processing and analyzing data - creating student reports. Coach 6 Studio MV is delivered with a wide range of exemplary Activities (placed in C:\Program Files\CMA\Coach6StudioMV\Full\CMA Coach Projects) which are described in this book. They serve as practical ideas of using Coach 6 Studio MV and illustrate features of the program. The Coach Activities can be used unchanged or Author users (teachers and educational material developers) can modify them to fit their educational needs. All modifications have to be done in the Author mode of Coach 6 Studio MV. The detailed description of all tools and options of Coach 6 Studio MV can be found in the manual Guide to Coach 6 Studio MV delivered with Coach 6 Studio MV. More examples of Coach Activities can be downloaded from the CMA site at http://www.cma.science.uva.nl. 4 I. Data Video Activities Table of Contents 1. INTRODUCTION...................................................................................................................................................... 6 2. INTRODUCTORY DATA VIDEO ACTIVITIES ............................................................................................................ 7 Activity: The start of the sprinter......................................................................................................................... 7 Activity: The high jumper .................................................................................................................................. 10 Activity: Video-yo (point tracking) .................................................................................................................... 15 Activity: Snooker shot........................................................................................................................................ 17 Activity: Capturing your own video................................................................................................................... 18 3. DATA VIDEO EXAMPLES ..................................................................................................................................... 20 Activity: Motion of a bicycle.............................................................................................................................. 20 Activity: Motion of two cars .............................................................................................................................. 21 Activity: Hitting a softball ................................................................................................................................. 22 Activity: Trampoline.......................................................................................................................................... 23 Activity: Basketball shot .................................................................................................................................... 24 Activity: Car collision........................................................................................................................................ 25 4. INTRODUCTORY DATA IMAGE ACTIVITIES .......................................................................................................... 26 Activity: Bixby Creek Bridge ............................................................................................................................. 26 Activity: Rectified Clifton Suspension Bridge.................................................................................................... 27 Activity: Analyzing motion with photography ................................................................................................... 28 5. DATA IMAGE EXAMPLES ..................................................................................................................................... 30 Activity: Zeeburger Bridge ................................................................................................................................ 30 Activity: Weights................................................................................................................................................ 31 Activity: Golden Gate Bridge ............................................................................................................................ 33 5 1. Introduction The Data Video Activities are used to make measurements on digital video clips or digital images. In these activities students are able to consider events, which happen outside the classroom. The events can be rather ordinary, every day events such as basketball shots, soccer kicks, amusement-park rides, plant growth or more unusual like jump on the Moon, car crashes or the motion of a manikin as it strikes an air bag during a car collision. During the video measurements the data points can be collected: - manually by clicking in selected frames on (a point of) a moving object, or - automatically by tracking (a point of) a moving object. From the frame rate with which the video clip was recorded, time information is deduced, while position information is measured on the two dimensions of the video image (picture), after calibration. During measurements on single images, position data or position and time data for stroboscopic images, are collected by clicking points of interest in an image. The collected data can be plotted in a graph, viewed in a table, used for further analysis and modeling and can even serve to calculate the locations of other points such as center of mass. Graphs are synchronized with the video frames. When scanning the data in the diagram the corresponding video frames are shown. This helps students to bridge the gap between the concrete visual display of a motion event and its abstract graphical representation. In Data Video Activities ready-to-go videos can be used or students can capture their own videos (e.g. using a simple web camera) and directly analyze them in Coach. Exemplary Data Video Activities are available in the Coach 6 Data Video project (icon). These Activities (in the Student user mode) are available also via CMA Coach 6 > Data Video icon of the Windows’ Start menu. Movies in Activities: 'Motion of bicycle', Motion of two cars' and 'Basketball shot' origin from the Video Clips Collection of Physics Education Group of Munich University and are used with special permission. Movies in Aactivities 'The start of the sprinter', 'The high jumper', 'Hitting a softball' and 'Trampoline' origin from the Video Disk Physic of Sports ¤ by D.A. Zollman and M.L. Noble, Kansas State University, available from Video Discovery Inc., and are used with special permission. Movies in Activities 'Car collision', origins from the Video Disk Physics and Automobile Collisions ¤ by D.A. Zollman, John Wiley and Sons, Inc. , available from Ztek Co., and are used with special permission. 6 2. Introductory Data Video Activities Activity: The start of the sprinter In this activity you will analyze in detail the motion of the sprinter during the beginning of the race. You will record the positions of the sprinter with time. Then you should deduce from your measurement how the speed and acceleration of the sprinter has changed with time. This example is used to teach you how to make measurements in the Data Video Activity. Instructions in the video measurement procedure are very detailed and will help you to work through the example.1 Video measurement procedure x Open Coach 6 Activity Introduction to Data Video > The start of the sprinter. Opening video x To open the video right click the Data-Video window and select Open… > Video. x Select The start of the sprinter video. x A video screen showing the first frame of the movie appears on the screen. Playing video x The video clip shows the start of the runner. x Play the video by clicking on the Play button on the video control bar. x To browse through frames use the frame controller. x By clicking frame on the frame controller a selected frame appears on the screen. At the right side of the frame controller you can find the Zoom button which allows zooming a part of the frame controller. First you have to select the frames you would like to zoom: click the first frame, while keeping the <Shift>-key pressed, click the last frame of the selection. Now click the Zoom button to zoom the frame selection. Video calibration Distance To perform the measurement you need to scale the video - you have to tell which distance on the screen corresponds to which actual distance. The "real life" distance of one-meter (white ruler) is shown on the screen. You have to also tell what your coordinate system is. x Right click the Data Video window and select the Change scale... option. x Since the vertical and horizontal meter sticks on the screen are the same length choose Same scale on all directions. 1 The picture by Vandy-Loubat-Petit/Agence Vandystadt/Photo Researchers, Inc origins from "Sprinting Start," Microsoft® Encarta® 97 Encyclopedia. © 1993-1996 Microsoft Corporation. All rights reserved 7 x x x x x The horizontal ruler (default red) and the coordinate system (default yellow) appear on the video screen. Move and match the horizontal red ruler with the horizontal white ruler. Specify in the Scale Settings dialog the scale length of 1 m. Position the coordinate system by dragging its origin (drag the small circle). (You can also rotate the coordinate system by dragging the yellow dot next to the origin). Click OK when you are ready. Time This option is used to specify how fast the video was taken. x Right click the Data-Video window and select the Time calibration... option. x This video clip was recorded with 30 frames per second. This information is used to connect the frame number with the time t (in seconds), as soon as you have decided which frame corresponds with t=0. Select the choice of t=0 at first selected frame. x Click OK when you are ready. Coordinate system settings x Right click the Data-Video window and select the Co-ordinate system... option. x Since the camera didn't move during recording of this movie choose Same at all frames. x Click OK when you are ready. Video Points Video points are points collected during the video measurement. x From the Tool menu select the Video Points... option. x You are going to measure only one video point so choose 1 measured point per frame. x Go to the Markers and colors part of the dialog window and choose your favorite color and marker for the video points (a white circle works fine). Frames You still have to tell how you want take a video measurement. Usually the manual video measurement is performed only on a number of selected frames. There are four possibilities to select frames for measurement. x From the Tool menu select the Frames… option. x There are four possibilities to select frames for measurement. For this example select Equally divided 20 frames between 1 and 74 frames. In this case it means you will measure circa every 4th frame. x All selected frames on the frame controller are marked black. Collecting data x Start the measurement by clicking the green Start button. x Coach automatically brings the first selected video frame on the screen. The cursor changes into . x Move the cursor (viewfinder) over the video screen to locate the video point for example a (white) shoulder or a head of the sprinter. x Click to store the first video point. x The video clip automatically advances to the next frame. Click the next point. 8 x Continue clicking on the location in each selected frame until you reach the last frame of the movie. Displaying data x To graph the data select the Display as a Diagram option and click the lower, left empty pane. x This will plot positions PX and PY versus time. This can be also done before you have started your measurement. The advantage of preparing the graph before is that you will see data appears on the graph during the measurements. x To see data in a table select the Tool menu option Display as a Table and click the lower, left pane. x One of the greatest features of the Data Video analysis is the ability to replay the situation and watch the graphs while movie plays. Click on the Replay button. Specify the Replay time or set the replay playing speed. Click OK. Your data should now be replaying on screen. You see live update of graphs and tables during the motion of the runner. Live updating If you are not satisfied with some of your marked video points you can come back to a frame and move a video point to another location. You do this by selecting the frame in the frame control bar and dragging the point to the desired location. Watch the diagram/table update simultaneously. (Note the sensitivity of the velocity plot to the location of the point.) You can also add some extra points after your measurement is finished. x Select extra frames by clicking a frame on the frame controller and press the <INS>- key on your keyboard. The selected frame becomes black. (You can deselect frames using the <DEL>- key). x Click the green Start button. The video automatically brings you to the selected frame. Locate your new video point. x Do the same for other extra selected frames. Assignments You are going to analyze the motion of the sprinter during the beginning of the race. x In the diagram you see P1X and P1Y versus time. To analyze the motion of the runner the horizontal position P1X is interesting (the vertical P1Y has changed very little during the motion). x Prepare the graph of P1X versus time. (Create/Edit diagram, for Column 3 make connection Empty). x Determine the greatest speed, at what time did the sprinter reach this speed? By determining the slope in the position versus time graph at a given point you will get the speed at that point (option Process/Analyze > Slope). x Create a velocity versus time graph (the speed is the derivative of the position!). Does the shape of this graph agree with your expectations? 9 Activity: The high jumper The aim in high jumping is to clear a crossbar resting at progressively greater heights between two upright standards. Most jumpers today employ the style known as the Fosbury Flop to clear the bar. The Fosbury Flop was named for its originator, American jumper Dick Fosbury, who used the style to win the event in the 1968 Olympics. American Dick Fosbury revolutionized the sport of To execute the flop, jumpers approach the high jumping. (All sport Historical)1. crossbar nearly straight on; they leap and twist on takeoff, rise above the bar headfirst, clear the bar with their backs oriented toward the ground, and land on the foam pad with their shoulders.1 By using Data Video, study in motion of a high jumper whom use the Fosbury technique can be done. To simplify this measurement, visual models of human motion can be used.2 The model is a variation on one used in the field of biomechanics. The human body as a stick figure; each stick or segment is treated as a rigid object with uniform density. The segments are connected by hinges, which are massless and are able to bend in appropriate ways. The mass of each segment is known from cadavers study. For this activity we consider three models: Model 1 The body is represented by a point mass located on hip. 1 "Track and Field," Microsoft® Encarta® 97 Encyclopedia. © 1993-1996 Microsoft Corporation. All rights reserved. 2 The idea of this lessons come from Physics and Sport by M. Larry Noble and Dean Zollman (Kansas State University), Video discovery, Seattle Wa 1989 10 Model 2 The body is represented by three rigid segments; one for the legs, one for torso and head, and one for the arms. Bending can occur only at the hip and the connection between arms and torso. Model 3 The body is represented by five segments as shown in the figure3. In this activity you will collect position data of the jumper. Then by using data about mass distribution and assuming that each body segment has a uniform mass distribution, you will calculate the location of the center of mass for different models. This example is used to teach you how to calculate and display on the video the center of mass of athlete. Model 1 x Open Coach 6 Activity Introduction to Data Video > 2a. The high jumper - point mass. x In the Data-Video window you see the first frame of the video which shows the high jump. x Play the video by clicking on the Play button. x The video is already scaled: one-meter scale on the video screen is used, clip was recorded with 30 frames per second. Select the Change Scale... option in the Tool menu to do the scaling. x Diagrams of the horizontal and vertical position are already displayed on the screen. x Start the measurement by clicking the green Start button. x Move the cursor over the video screen and click to locate the video point. Mark the position of the jumper's hip. x The video clip automatically advances to the next frame. Click the next point. Continue clicking on the location in each selected frame until you reach the last frame of the movie. x Described the resulting graphs. x Save your results. You will need them in the next activity. 3 All pictures of different models origin from videodisc 'Physics and Sport' by M. Larry Noble and Dean Zollman (Kansas State University), Video discovery, Seattle Wa 1989 11 Model 2 Now you are going to use the three-segment model. For this model the body is represented by three rigid segments; one for the legs, one for torso and head, and one for the arms. The three segments have different masses and account for different percentages of the total center of mass of the jumper (look at the picture Model 2). The legs are 32.7% of the total body mass, the torso (incl. the head) is 57.5% and the arms are 9.8%. So the coordinates of the center of mass of the jumper for this model are: xc = 0.327*x1+0.575*x2+0.098*x3 yc = 0.327*y1+0.575*y2+0.098*y3 where x1, y1 are coordinates of the legs segment x2, y2 are coordinates of the torso segment x3, y4 are coordinates of the arms segment x Open Coach 6 Activity Introduction to Data Video > The high jumper - 3 segment model. x In the Data-Video window you see again the movie of the high jumping. x The video in the Data Video window is already scaled: one-meter scale on the video screen is used, film was recorded with 30 frames per second. Select the Change scale... option in the Tool menu to see the scaling. Video Points x In this activity you are going to measure three video points. x Right click the Data-Video window and select the Video Points… option, 3 measured points are already chosen. Displaying data in diagrams x You are going to create two diagrams Horizontal position versus time and Vertical position versus time. x Create the Horizontal position diagram. - Click the yellow Diagram button and click the New diagram button. - Give a diagram name: Horizontal position. - Choose for Column 1 as connection Clock, give a quantity name: time and unit: s. - Choose for Column 2 as connection P1 - X, give a quantity name: x1 and unit: m. - Choose for Column 3 as connection P2 - X, give a quantity name: x2 and unit: m. - Choose for Column 4 as connection P3 - X, give a quantity name: x3 and unit: m. - Choose for Column 5 as connection Formula, give a formula: 0.327*x1+0.575*x2+0.098*x3 and give quantity name: xc and unit: m. This will calculate x coordinate for the center of mass. x Create the Vertical position diagram. - Click the yellow Diagram button and click the New diagram button. - Give a diagram name: Vertical position - Choose for Column 1 as connection Clock, give a quantity name: time and unit: s. - Choose for Column 2 as connection P1 - Y, give a quantity name: y1 and unit: m. - Choose for Column 3 as connection P2 - Y, give a quantity name: y2 and unit: m. - Choose for Column 4 as connection P3 - Y, give a quantity name: y3 and unit: m. - Choose for Column 5 as connection Formula, give a formula: 0.327*y1+0.575*y2+0.098*y3 and give quantity name: yc and unit: m. This will calculate y coordinate for the center of mass. x Place the diagrams in panes. 12 Displaying the center of mass point on the video screen 1. Select the Tool menu Video Points... option. 2. Choose a number of calculated points - 1. 3. Choose for X coordinate of the calculated point: Formula xc. 4. Choose for Y coordinate of the calculated point: Formula yc. 5. This will plot the calculated point - the center of mass for the jumper - on the video screen during the measurement. Collecting data During the measurements you have to click three location on every selected frame. You must choose midpoints of each of the three model segments. First try to imagine three segments for the body. For the leg segment: since the feet may be widely separated, you must choose a point midway between them for one end point of the legs segment. The other endpoint should be at a point midway between the hips. For the arm segment: this segment starts at a midpoint between the hands and end midway between the shoulders. For the torso-head segment: this segment runs from the top of the head to the bottom of the trunk. The measurement points for three- x Start the measurement by clicking the green Start segment model are the midpoints of each of the three segments. button. x Move the cursor over the video screen and click on the video screen to locate the video points. Mark the midpoint of each of the three segments. (See the picture above). x The calculated point appears shortly on the movie screen and the movie automatically advances to the next frame. Click the next point. x Continue clicking on the location in each selected frame until you reach the last frame of the movie. x Data appears in the diagrams. Tip: If you want to display on the diagram only coordinates of the center of mass make other axes invisible. x One of the greatest features of the data video analysis is the ability to replay the situation and watch the graphs while movie plays. Click on the Replay button. Specify the replay time or set the replay playing speed. Click OK. Your data should now be replaying on screen. You see live update of graphs and tables during the motion of the athlete. Model 3 To calculate the center of mass you can use also the more complex five-segment model. x Open Coach 6 Activity Introduction to Data Video > The high jumper - 5 segment model. x In the Data-Video window you see again the same movie. x The video is already scaled: one-meter scale on the video screen is used; film was recorded with 30 frames per second. Select the Change scale... option to see how it is done. x The diagrams are already prepared for you. 13 x Start the measurement by clicking the green Start button. x Move the cursor over the video screen and click to locate the video points. Mark the midpoint of each of the five segments (look at the picture Model 3). x The calculated point - the center of mass - appears shortly on the screen and the video automatically advances the next frame. Click the next point. x Continue clicking on the location in each selected frame until you reach the last frame of the video. x Data appears in the diagrams. Assignments x Is the center of mass of a high jumper always inside the body of the jumper? Because the body configuration of a high jumper changes, the location of the center of mass relative top the body can also change. The three-segment model provides a more accurate determination of the center of mass then the fixed center of mass model. x When high jumper is directly over the bar, is his center of mass located on his body or below? x Watch the frames as the jumper clears the bar. Is there any time when his center of mass is at the center of his body? If not, where is it relative to his body? x Could a high jumper clear the bar but have the center of mass under the bar? x Analyze the high jump in terms of energy, you can assume that the jumper's gravitational potential energy as if the entire mass is concentrated at the center of mass. Compare, in terms of energy, a Fosbury flop as shown on the movie and straight jump where the entire body is above the bar. x Can you explain now why jumpers who use the Fosbury flop have successfully cleared the bars at greater heights than many that do not use it? Compare results of different models (use the Import background graph option). x Is the increase in accuracy worth the increase in the complexity of the model? 14 Activity: Video-yo (point tracking) In this activity you will analyze in detail the motion of a giant yoyo. You will collect the position data of the moving point located on the yo-yo. This example is used to teach you how to use point tracking the automatic mode of collecting video points. Video measurement procedure x Open Coach 6 Activity Introduction to Data Video > Video-yo (point tracking). Playing video x Play the video by clicking on the Play button. x You see a giant winding yoyo. You will use point tracking to record the motion of a point near the rim. This point is marked by sticker. Video calibration x Right click the Data Video window and select Change Scale.... x Because the vertical and horizontal scales are the same, select Same scale in all directions. x Drag and resize the end points of the scale-ruler (red by default) to measure the diameter of the yoyo. Specify the scale length of 0.34 m. x Position the origin of the coordinate system at the chair seat the person is standing on. x Click OK. x Right click in the Data Video window and select Time calibration.... The video clip has been recorded with a frame rate of 30 frames per second. x Select t=0 at first selected frame and click OK. Video points x Right click the Data Video window and select Video Points.... x You are going to measure 1 point per frame, viz., the point near the rim of the disk. Leave the current settings of Number of measured points per frame. x Go to the Markers and colors part of the dialog window and choose your favorite color and marker for the video point (a blue big dot works fine). Frames x Move the Begin and End markers (blue triangles) in the frame controller to ignore useless frames at the beginning or end of the video clip. x Right click the Data Video window and select Frames... x For point tracking you can easily select all frames for measurement. Select Use All frames. x Click OK. All selected frames are marked black in the frame controller. Collecting data by Point tracking x Right click the Data-Video window and select the Point Tracking option. x Coach automatically brings the first selected frame (tracking frame) in view (indicated as red frame on the frame controller). x In the tracking frame you see the tracking area and the search area. Move the tracking area (P1) to the appropriate location (the black point on the yo-yo). 15 x x x To plot the data points in a diagram right click the Data Video window and select Display as a Diagram.... The shape of the cursor changes into a small diagram. Click the left, lower pane to place the diagram. Start measuring by pressing the green Start button. Coach starts to collect data. You can stop measurement at any moment by clicking the Stop button. If point tracking fails at some moment you can adjust measurement (see the detailed description below). During the measurement the hand of the person holding the yo-yo is moving. You can improve the accuracy of your measurement by using the moving co-ordinate system. x Right click the Data Video and select Erase all points and remove the first collected points. x Right click and select Co-ordinate system. Select as Origin the First point clicked at each frame. x Right click and select Point Tracking Settings. Notice that the Tracking area is now defined as well for Video Point P1 as for the Origin of the Co-ordinate system. x Right click and select Go to Tracking Frame. Move the tracking area of the Origin to the hand location. x To adjust the diagram click the Zoom to fit button in the Diagram pane. x Start the measurement again. x Deselect the Point Tracking option if you do not want to see the tracking areas anymore. Adjusting measurement The automatic point tracking can fails on some frames. Once the measurement is finished you can drag erroneous measured points to correct positions. You do this by selecting the frame in the frame controller and dragging the point to the desired location. The point tracking can also fails in the range of frames. In such a case it is possible to repeat the measurement only in a selection of frames where tracking was not correct. x First select the frames on which the tracking failed. Click the frame where selection begins. While pressing the <Shift>-key click the last frame of the selection. x Remove data associated with these frames by using the <Backspace>-key. x Now the first selected frame without associated data is chosen by Coach as the new tracking frame. If needed change the point tracking settings. x Start the measurement again to continue point tracking from this frame on using the new settings and measure all selected frames without data. x It is also possible to measure the remaining frames manually. Assignments x In the diagram, the horizontal (P1X) and vertical (P1Y) Cartesian coordinates of the video points are plotted against time. Select the Analyze /Process > Function-fit menu item to find regression curves for average values of P1X and P1Y (use sine fit and quadratic fit, respectively). x Make a new diagram to plot the vertical (P1Y) Cartesian coordinate against the square of time. Use Function Fit to find a regression curve as a sum of a straight line and a sinusoid. 16 Activity: Snooker shot In this activity you will analyze the motion of the moving ball after a snooker shot. This is an example of the advanced video measurement. The measurement is a combination of the manual measurement and point tracking. Additionally the perspective correction is used to correct the distorted video. Video measurement procedure x Open Coach 6 Activity Introduction to Data Video > Snooker shot. Correction of the perspective distortion x Right click the Data-Video window and select the menu option Perspective correction. A red perspective rectangle is placed on the top of the video screen. x Drag the corners of the rectangle one by one to the corners of the billiard table. Every time a corner of the perspective rectangle is moved the video screen is distorted. After mapping all four points the distorted billiard table is transformed into corrected rectangle. x Now you may deselect the menu item Perspective correction. Calibration x Right click the Data-Video window and select the Change Scale... option. x Select Different scale in horizontal and vertical direction. x Move horizontal and vertical scale-rulers such that they match the sizes of the rectangle (table). x Fill out the scale lengths: height of the billiard table is 3.6 m and its width is 1.8 m. x Move the origin of the coordinate system to a convenient position, e.g. the center of the billiard table. x Click OK to close the Change Scale dialog. Collecting data x In the video the prediction of the white ball's trajectory is given with the while line. Because of this it is not possible to use the point tracking for the measurement at the first part of the video. This part of the measurement has to be done manually. When the white trajectory line disappears the Point Tracking option can be turned on. x Press the green Start button. x Click on the cue ball; it is less visible because of the white overlay. x Once the ball is far enough away from the overlay, stop manual measurement by pressing the Stop button. x Select the menu item Point Tracking and drag the tracking area over the cue ball. x Press the green Start button again to continue data collection by point tracking. In the diagram you get the graphs of the recorded coordinates versus time and the graph of the velocity of the cue ball versus time. Do you understand the graph? Explain it. 17 Activity: Capturing your own video In this activity you will learn how to capture your own video clips. It is possible as well in the Data-Video window as in the Video pane (but not in both windows at the same time). In this activity capturing it is done via the Data Video window. Video capture is one of the most system-intensive tasks you can demand of a personal computer. Good results depend on the performance and capacity of all components of your system that have to transport frames from a video-capture device to the processor and hard disk. Frames will be dropped from the captured clip if one of these components cannot handle the workload. Make sure your system is optimized by reviewing and following the guidelines set by the manufacturers of your video-capture device and hard drive. Before you start capture your camera should be plugged in and turned on. When Coach does not “see the camera” then the Capture option is disabled. x Open Coach 6 Activity Introduction to Data Video > Capturing your own video. Starting Capture x Open the Data Video Capturing window by right-clicking the Data Video window and selecting the Capture option. The Data Video – Capturing window includes a preview area, which displays your currently recording video and control buttons (Record and Stop) for recording video. Capture settings Right click the Data Video – Capturing window and select the Capture settings option. x Specify the maximal time of capturing, for example 10 s. x Specify the video resolution (the height and width of the video clip in pixels) - keep 320x240 pixels. A larger frame size lets you see more detail but requires more processing. The lowest frame size that is acceptable in most situations is 240x180 pixels. x Set frame rate to 5 frames per second. x Click OK. Recording a video Use the Record button to start recording a video. The recording will stop automatically when the given maximal capture time has elapsed. Alternately, you can click the Stop button or <Esc>key to stop recording before the time expires. x By checking the Capture information option the information about the recorded video can be obtained. x You can view the recorded video by using the Show > Playback option. x The recorded video file is a temporary file (with default name Rec###.avi), and is replaced by a new file each time the new recoding is started. x To use the recorded video right click the Data Video – Capturing window and select Use recorded video. The video clip will be transferred into the Data Video window for editing and measuring on your video clip. Depending on a video file size it may take a few minutes. Removing the beginning and the ending of a video clip To remove several frames from the beginning and/or end of a video the Begin and End blue markers can be used. 18 x x Slide the Begin and End blue markers located on the frame controller to the desired first and last frame of your new video clip. The edited video can be previewed by clicking the Play button. Adjusting video After selecting the Adjust video… option you can: x change the video brightness; x change the video contrast; x rotate the video; x flip the video horizontally or vertically. All the video adjustments (filters) are applied in real-time on the video file, without writing a new file. These parameters are stored in the activity. By using the Export video option the modified file can be saved as a separate video file. Annotating a video The Annotations allows placing text labels on frame(s) of a video clip or on an image. x Right click in the Data-Video window and select Annotation. x Type in your text. x If desire click the Font button to change the text font and color. x Check the Framed option to place the annotation in a frame. x Check the Shown on all frames option to place the annotation on all video frames. When this option is not checked the annotation appears only on a frame which is actually displayed in the Video window. x Click OK. If necessary move the annotation to another location on the video screen. Exporting video The video clips captured in Coach become a part of Coach Activity when the activity is saved. To save a captured video file as a separate file on the computer hard drive use the Export video… option. To export a video file: x Right click in the Data-Video window and select Export video…. x Type in a file name. x Specify video file settings. Before you save in AVI format, you may choose to change Video compressor (and Audio compressor if your video clip includes audio track). x Click the Save button to store a video on your hard disc. Depending on the video file size it may take a few minutes to save the file. 19 3. Data Video Examples Activity: Motion of a bicycle Suppose you go on your bicycle for a ride. When you pedal hard to gain speed on your bicycle, your velocity changes. It changes again when you slow down to stop. What would be your acceleration during such situations? In this activity you are going to investigate a motion of a bicycle during speeding up and slowing down. Activity procedure 1. Open Coach 6 Activity Data Video Examples > Motion of a bicycle. 2. In the Data Video window you see the first frame of the video Start of the bicycle. 3. Play the video. Describe the bicycle's motion. 4. Measure the position of the cyclist (choose as a video point location for example an axis of the bike. 5. The video is already scaled (1.72 horizontal ruler on the movie screen, speed of 25 frames per second). 6. The horizontal position versus time graph of your measurements appears on the screen. Assignments Speeding up bicycle x Describe the motion of the cyclist. - What was the initial position of the cyclist? - What was the final position of the cyclist? - How long was the motion? - What can you deduce about the cyclist's speed? x Create a graph of the horizontal velocity versus time (use the 'Derivative' option). - What was the initial velocity of the cyclist? - What was the final velocity of the cyclist? - What can you deduce about the cyclist's acceleration? x Calculate the average acceleration of the cyclist. You can do it in different ways by: - by calculating the change in velocity divided by the corresponding change in time, - using the quadratic curve fit for position data (go to Process/Analyze > Function fit). Slowing down bicycle Now you are going to analyze the bicycle's motion as it comes to a stop. x Right click the Data Video window and open the video Stop of the bicycle. x Click the Coordinate Settings… option and scale the video by using 1.71 m horizontal ruler on the video screen. Set the video speed to 25 frames per second. x Click Frames and select the frames to measure. Check Individual frames and type 1-47$2, which means that every 2nd frame, will be selected. x Describe the bicycle's motion and answer question from the Speeding up bicycle section. 20 Activity: Motion of two cars Two cars are coming on the crossing. In this activity you will investigate the cars’ motions. Activity procedure 1. Open Coach 6 Activity Data Video Examples > Motion of two cars. 2. Play the video. Explain what is happened. 3. The video is already scaled (4.60 horizontal ruler on the first movie screen, speed of 25 frames per second). 4. Measure the positions of white and black cars (choose easy to recognize points). Remember that at every frame you have to first click a position of the first car (for example the white one) and then click a position of the second car (for example the black one) then the program brings the next frame to measure. 5. The horizontal position versus time graph of your measurements will appear on the screen. Positions of both cars will be plotted. Assignments x Describe the motion of the white car. - Is the car moving in the direction of increasing of decreasing distance from the origin. - What was the initial position of the white car? - What was the final position of the white car? - How long was the motion? - What can you deduce about the white car's speed? x Describe the motion of the black car. - Is the car moving in the direction of increasing of decreasing distance from the origin? - What was the initial position of the black car? - What was the final position of the black car? - How long was the motion? - What can you deduce about the black car's speed? x Create a graph of the horizontal velocity versus time for both cars (use the 'Derivative' option). - What was the initial velocity and final velocity of the white car? - What was the initial velocity and final velocity of the black car? - Explain why the velocity of the black car is negative? - What can you deduce about the acceleration of the white car? - What can you deduce about the acceleration of the black car? x Calculate the average acceleration for both cars. 21 Activity: Hitting a softball In this activity you are going to measure (by point tracking) on the video showing a player hitting the softball. You will calculate the impulse experienced by the ball when it is hit with the bat. The mass of the ball is given: m = 0.192 kg. Which two speeds do you need to know in order to be able to calculate the impulse on the ball? Having found the impulse on the ball, you can also try to calculate the force exerted by the bat on the ball during their collision. Activity procedure 1. Open Coach 6 Activity Data Video Examples > Hitting a softball. 2. In the Data Video window you see the first frame of the video. Play the video. 3. The video was recorded with very high-speed of 454 frames per second. Because of the high recording speed you can observe the details of the motion more closely. 4. The video is already scaled and the graphs are prepared. 5. Right click the Data Video window and select Point Tracking. 6. Coach automatically brings the first selected frame, the Tracking frame, in view (the red frame on the Frame controller). In the tracking frame you see the tracking area and the search area. Move the tracking area (P1) to the appropriate location (ball). 7. Start measuring by pressing the green Start button. 8. After the measurement is finished you can drag erroneous measured points to correct positions (see frames 23, 24, 25, 26). 9. Deselect the Point Tracking option if you do not want to see the tracking areas anymore in each frame. 10. Examine the graphs of your measurements. Assignments x Determine the speed of the ball before and after the shot. (Find the slopes of the first and last parts of your graph by using the 'Slope' option). x Calculate the impulse on the ball. x To calculate the force on the ball (during the collision) you need to know the contact time of ball and bat. You might be able to estimate that by looking at consecutive frames of the film, and by counting the number of frames where there was contact. (There can't be many!) Try and determine the contact time from the movie. Use the frame bar under the movie. In how many frames is there contact? Can you determine the contact time from this? (Use the data table.) x What can you conclude from this about the force that the bat exerted on the ball, during the collision? 22 Activity: Trampoline In this activity you will investigate the movement of a trampoline jumper; you will concentrate on the analysis of the vertical movement of the athlete.4 Activity procedure 1. Open Coach 6 Activity Data Video Examples > Trampoline. 2. In the Data Video window you see the first frame of the video clip. Play the video. Explain the athlete's motion. 3. The video is already scaled and all frames are selected for measurement. 4. You are going to use point tracking. Right click the Data Video window and select Point tracking. 5. Move the tracking area (P1) on the tracking frame to the appropriate location (the jumper's black short). 6. Start measuring by pressing the green Start button. The collected data appears on the graph of the vertical position of the athlete versus time. 7. Deselect the Point Tracking option if you do not want to see the tracking areas anymore in each frame. Assignments x Look at the graph carefully; does it have a pure sinusoidal form? When is the speed of the athlete zero? Where is that in the motion? x Prepare the velocity versus time v vs. t graph (use the Derivative option). x The movement of the athlete can be divided into the following portions: (a) the athlete moves upward through the air. (b) the athlete moves down through the air. (c) the athlete, in contact with the trampoline moves down. (d) the athlete, in contact with the trampoline moves up. (e) the athlete makes somersault. Show these sections, (a) (b) (c), (d) and (e) in the v vs. t graph. x Prepare the acceleration versus time a vs. t graph. Show the sections (a), (b), (c) (d) and (e) in the a vs. t graph as well. Give an explanation of the different parts of the diagram. x What is the lowest (ie. most negative) value of a on the graph? What is highest positive value of a on the graph? x How would you describe this athlete's motion? x Analyze the trampoline jump in terms of energy. The picture by John Eastcott/YVA Momatiuk/The Image Works origins from "Trampoline," Microsoft® Encarta® 97 Encyclopedia. © 1993-1996 Microsoft Corporation. All rights reserved. 4 23 Activity: Basketball shot The motion of the basketball is an example of projectile motion. The motion can be divided into two parts: the horizontal component and vertical component. These two components can be calculated independently for each other and then the results can be recombined to describe the total motion. After the basketball shot the only forces acting on the ball are the forces of gravity and air resistance. The force of air resistance depends on the ball's velocity and its mass-tocross sectional area ration. As the velocity of the ball increases the force of air resistance increases, while as the mass-to-cross sectional ration increases the air resistance decreases. Is the force air resistance considerable and plays a role in determining the ball's trajectory? In this activity you will try to answer this question. Activity procedure 1. Open Coach 6 Data Video Examples > Basketball shot. 2. In the Data Video window you see the first frame of the video. Play the video. Explain the ball's motion. 3. The video is already scaled. Start your measurement by clicking the green Start button and measure the position of the ball. 4. The horizontal and vertical positions of the ball appear in the diagram Position of the ball on the screen. 5. Examine the graphs. Assignments x Explain the graph. Describe the motion in horizontal and vertical direction. x Determine the initial velocities of horizontal and vertical motion. x Create a horizontal velocity versus time graph and a vertical velocity versus time (the speed is the derivative of the position!). x Explain both graphs. x Determine the acceleration at the beginning and at the end of the motion. By determining the slope in the velocity versus time graph at a given point you will get the acceleration at that point (option Process/Analyze > Slope). x Does acceleration change during the motion? Can you explain why? x Is the effect of air resistance significant? 24 Activity: Car collision A collision is a process involving two objects, each of which exerts a force on the other. The car crash shown on the video in this activity is an example of inelastic collision. In general, the total momentum remains constant for a system of objects that interact with one another. In this case, the automobiles are objects which interacting so the total momentum should remain the same during their motion. What happened with the kinetic energy of the cars during collision? In this activity you will investigate these problems. Activity procedure 1. Open Coach 6 Data Video Examples > Automobile collision. 2. In the Data Video window you see the first frame of the video. Play the video. 3. You can also see video of the same collision taken from a different angle. Click the Display video button and place the video in one of the panes. 4. Describe the motion of the automobiles. 5. The video was recorded with very high-speed of 786 frames per second. Because of the high recording speed you can observe the details of the motion more closely. 6. The video is already scaled and the graphs are prepared. 7. Carry out measurements on the position of the cars (mark the point at the end of black stips on the cars). 8. Examine the graph of your measurements. Assignment x On the screen you see the two graphs: Position of the first car and Position of the second car. x Create for both cars graphs of the horizontal px= m* vx and vertical momentum py=m*Vy where vx is the horizontal component of the car velocity and vy is the vertical component of the car velocity. Both cars have the same mass of 800kg. x Create the graph of total momentum of the two cars system. Remember that momentum is a vector quantity! First calculate the total horizontal momentum px, then total vertical momentum py and final total momentum by: p px2 py2 x Is momentum conserved during the automobile collision? Are these cars isolated objects? Is there friction force acting? x Create graphs of Kinetic energy of the first car and the Kinetic energy of the second car. To 2 2 calculate the kinetic energy use the following formula: Ek 1/ 2 * mvx 1/ 2 * mvy . x Create the graph of total kinetic energy of the two cars system. Kinetic energy is not a vector quantity! Is the kinetic energy conserved during this collision? 25 4. Introductory Data Image Activities Activity: Bixby Creek Bridge In this activity you will analyze the shape of Bixby Creek Bridge in California. This example is used to teach you how to make Data Image measurements. Activity procedure x Open Coach 6 Data Video Introduction > Bixby Creek Bridge. Opening an image x To open an image right click the DataVideo window and select the Open... > Single picture option. x Select Bixby Creek Bridge in California. x A screen showing a photograph of Bixby Creek Bridge in California (highway 1, 13.3 miles south of Carmel) appears on the screen. The central span of this bridge is 320 feet long and 265 feet high. This span is supported by an arch underneath. Calibration x Right click the Data Video window and select Change Scale.... x Select Same scale in all directions. x Move the scale-ruler such that it matches the roadway between two main piers supporting bridge. x Fill out the scale length: 100 m. x Move the origin of the coordinate system to the middle point of the bridge. Video points x Video points are points collected during the measurement. Right click the Data Video window and select the Video Points... option. x Set the Number of measured points to 12. Collecting data x Press the green Start button. x Click on suitable points on the supporting beam of the bridge, starting from left and continue to right. x The measurement will stop automatically after 12 video points have been collected. x If you are not satisfied with some of your marked points you can select a point, it becomes active and changes its color, and move it to another location. Watch the diagram update simultaneously. 26 Displaying of data x To plot the data right click in the Data Video window and select Display as a Diagram. The shape of the cursor changes into a small diagram. Click the lower left pane to place the diagram. x In the diagram, the horizontal (P1X) and vertical (P1Y) Cartesian coordinates of the data points are plotted against each other. If desired, you can change the names P1X and P1Y into X and Y, respectively. Analysis In the diagram, the data points seem to lie on a parabola. Right click the diagram and select Process/Analyze > Function Fit and try to find the parabola that fits best. What do you think of this approach? Activity: Rectified Clifton Suspension Bridge In this activity you will analyze the shape of Clifton Suspension Bridge and try to answer the question “Is the shape of the arch of this bridge a parabola?” This example is used to teach you how to use the Perspective Correction option on digital images. Activity procedure x Open Coach 6 Data Video Introduction > Rectified Clifton Suspension Bridge. x The image is already scaled: the roadway between the piers is 214 m long and the height above roadway of the pier in the background is 26.2 m. The origin of the coordinate system is located on the lower left corner of the pier in the background. x Press the Start button. x Click on suitable points on the front arch of the bridge, start at left and continue to right. x If you are of opinion that you have collected sufficient data, press the Stop button. In the diagram, the vertical (P1Y) and the horizontal (P1X) Cartesian coordinates of the data points are plotted against each other. If desired, you can change the names P1X and P1Y into X and Y, respectively. x In the diagram, the data points seem to lie on a parabola. Is this true? x Right click the Diagram pane and select Process/Analyze > Function Fit and try to find the parabola that fits best. What do you think of this approach? x The measured positions on the cable do not lie on a parabola; the best quadratic fit does not match well with the data. With help of the Perspective correction option the plane formed by the piers and the left side of the road can be rectified to a fronto-parallel view. Then the cable is almost in this plane and a more realistic measurement can be done. x Before you start the measurement erase your video points by right clicking the Data-Video window and selecting Erase All Values. 27 Correction of the perspective distortion x Right click in the Data Video window and select the Perspective Correction option. A rectangle is placed on top of the picture. x Translate the rectangle such that its lower left corner matches the bottom of the pier in the background. Move the rest of the corners of the rectangle one by one to the bottom and top of the two piers. The image will be transformed accordingly. x If desired, you can scale by moving the cursor to a side of the rectangle so that the cursor becomes a two-sided arrow and by dragging the horizontal or vertical side. Image calibration x In the corrected, fronto-parallel view of the bridge, the scaling in horizontal and vertical direction is not the same. You have to re-scale your image. x Right click in the Data Video window and select the Change Scale... option. x Select in the dialog window Different scale in horizontal and vertical direction. x Move horizontal and vertical scale-rulers such that they match the sizes of the rectangle. x Fill out the scale lengths in the dialog window: height is 26.2 m and span size is 214 m. x Move the origin of the coordinate system to the lower left corner of the rectangle. x Now you may deselect the menu item Perspective Correction. Collection of data x Press the Start button. x Click from left to right on suitable points on the front arch of the bridge. x If you are of opinion that you have collected sufficient data, press the Stop button. Analysis x What shape has the Clifton Suspension Bridge? Is this a parabola? Activity: Analyzing motion with photography Stroboscopic photographs can be used to study motion qualitatively and quantitatively. In case of very rapid motions, like the motion of the falling ball, each time the strobe light flashes, an image of the ball is recorded. Successive strobe flashes are always separated by equal time intervals. Qualitatively, the spacing between images can be used to calculate velocities and accelerations. In this activity you will analyze the stroboscopic photograph with recorded images of a ball at 0.033 s intervals as it was being dropped. This example is used to teach you how to make measurements on stroboscopic pictures. Activity procedure x Open Coach 6 Data Video Introduction > Analyzing motion with photography. Distance calibration x Right click the Data Video window and select Change Scale.... x Select Same scale in all directions. x Move the scale-ruler such that it matches the 1 m part of the meter stick. x Fill out the scale length: 1 m. 28 x Move the origin of the coordinate system to the lowest ball image. Time calibration x Right click the Data Video window and select Time calibration.... x Select Stroboscopic picture. x Select Stroboscopic time interval and fill 0.033 s. Video points x Select the Video Points... option. x Set the Number of measured points to 17. Collecting data x Press the Start button. x Click ball images, starting from top and continuing to bottom. x The measurement will stop automatically after 17 points have been collected. x If you are not satisfied with some of your marked points you can select a point, it becomes active and changes its color, and move it to another location. Displaying of data x Right click the Data Video window and select Display as a Diagram to plot the data points. The shape of the cursor changes into a small diagram. Click in the lower right pane to place the diagram. x In the diagram, the horizontal (P1X) and vertical (P1Y) Cartesian coordinates of the data points are plotted against time. You are interested in the vertical motion of the ball and you can make the P1X axis invisible (use the Create/Edit diagram option). Analysis 1. Describe the motion of the ball. Describe the distances between images of the ball during its motion. What can you deduce about the ball's speed? 2. Create a graph of the vertical velocity versus time (use the Derivative option). x What was the initial velocity of the ball? x What was the final velocity of the ball? x What can you deduce about the ball's acceleration? 3. Calculate the average acceleration of the ball (use the Slope option). How does the measured acceleration compare with the gravity acceleration? 29 5. Data Image Examples5 Activity: Zeeburger Bridge In this activity you analyze the shape of the Zeeburger Bridge in Amsterdam across the Amsterdam-Rhine Canal. You record the coordinates of points on the arch bridge and you derive from these data the height as a function of the horizontal distance. Activity procedure 1. Open Coach 6 Data Video Examples > Zeeburger Bridge. 2. Calibrate your image by using a white ruler of 80 m length. x Right click the Data Video window and select Change scale.... x Drag and resize the end points of the scale-ruler (red by default) to match the horizontal white ruler. Specify the scale length of 80 m. x If desired, you can move the coordinate system by dragging its origin (drag the small circle). Locations for which the mathematical formulas will be simplest for symmetry reasons are the top of the arch or the points on the road straight under the top of the arch. x Press OK. 4. Start your measurement by pressing the Start button. 5. Click on suitable points on the front arch of the Zeeburger Bridge (from left to right). 6. The data appears in the diagram of the horizontal (P1X) and vertical (P1Y) Cartesian coordinates of the video points are plotted against time. 7. If you are of opinion that you have collected sufficient data, press the Stop button. Assignments Analyze the mathematical shape of one of the arches of the Zeeburger Bridge. In mathematical terms: what function has a graph that fits well with the arch and stays closest to the measured video points. x Make a diagram in which P1Y is plotted against P1X, if you have not done this yet, and change the names P1X and P1Y. If you replace in the Create/Edit diagram… dialog window the quantity P1X, connected with column C2, by x, and the quantity P1Y of column C3 by y, then you have “normal” mathematical symbols. x Right click in the Diagram window and select Scan. Place the cursor at a point in which you are interested and read off its values. You can also read off the values in the table. x Verify the answer by measuring the position with a computerized ruler. Right click the Data video window and select Show > Ruler. Drag the endpoints of the ruler so that they lay on the points between which you want to know the distance. 5 The first two activities included in this project are part of the Bridges and Arches project which has been developed by André Holleman and André Heck as part of research work in AMSTEL Institute. More information about this work can be found at: http://www.science.uva.nl/~heck/research 30 x In the diagram, the data points seem to lie on a parabola. Right click the Diagram window and select Process/Analyze > Function Fit and try to find the parabola that fits best. Activity: Weights The arch of the Zeeburger Bridge can be seen as a parabolic shape, on which hang equal weights at equal distances (in horizontal direction). Each beam “bears” a same portion of the road surface. The arch of the Creek Bridge (project Data Video Introduction > Bixby Creek Bridge Activity) has also a parabolic shape which support equal weight placed at equal distances form each other (again, equal distances in horizontal direction). In this activity you are going to study the “inverted” model: a weightless chain on which hang equal weights at equal horizontal distances. Activity procedure 1. Open Coach 6 Data Video Examples > Weights. 2. The image shows a chain with hanging weights. These weights have been placed symmetrically such that the horizontal distance is always 1 dm. 3. Coordinate system and scaling have already been chosen. 4. Click the Start button and measure seven video points: the two suspension points and the five points on the chain to which the five weights have been attached. The drawing below is a simplified display of the video image. Assignments 1. Measure in the video screen with the protractor the angles of inclination A, B, and C. Measure in the diagram window the slope of the segment AB, of segment BC, and of segment CD. Maybe you notice some regularity in the angles and/or slopes or in their ratios? 2. Draw on paper the graph of the standard parabola y = x2 with A(0,0), B(1,1), C(2,4), and D(3,9). Compute A, B and C. Compute the slope of AB, of BC, and of CD. 3. Investigate whether the regularity that you have found in exercise (2) also D occurs for the standard parabola. If you did not find regularity in (2), try if you can find the regularity of exercise (3) back in (2), approximately. (If desired, compute the coordinates of more points of the standard parabola.) 4. You have seen that the model of weights C hanging on a weightless chain indeed R leads to a parabolic shape. Hopefully you have also found that there exists a simple relation between the slopes of consecutive straight segments of the B chain. This regularity can be explained Q by equilibrium of forces acting on the P chain. A 31 The figure below is the basis of our reasoning: C FBs2 B FBs1 FAs FAs Fz = 1 A Fz = 1 The (equal) weights lead to a gravitational force Fz in each point of application. For simplicity it is assumed that Fz = 1. The weight A hangs in equilibrium by the tension forces in FAs in the left- and the rightsegment of the chain. Because of symmetry the tension forces on the left and on the right are equal in magnitude. Easy reasoning gives that the vertical component of FAs must be equal to 0.5Fz. A gravitational force and two tension forces cause the equilibrium of the point of application B. x Reason out that the slopes of AB, BC, and CD (if another weight would be placed) are in ratio 1 : 3 : 5 . 5. Not every hanging chain has a parabolic shape. More strongly: Christiaan Huygens (16291695) proved that an unloaded homogeneous chain certainly not hangs in a parabolic shape. Such a freely hanging chain is displayed on the image Necklace. x Open the Necklace image and try to find the shape of a perfectly flexible chain hanging under gravity? 32 Activity: Golden Gate Bridge In this activity you analyze the shape of the Golden Gate Bridge. Activity procedure 1. Open Coach 6 Data Video Examples > Golden Gate Bridge. 2. The image shows the Golden Gate Bridge. 3. Correct the perspective distortion. x Right click in the Data Video window and select Perspective Correction. x Translate the rectangle such that its lower left corner matches the left bottom of the tower at the roadway. Drag other corners of the rectangle one by one to the bottom at the roadway and top of the two towers. The image will be transformed accordingly. 4. Calibrate the image. Use the following data: the height of the towers above the roadway is 152 m and the size of the main span is 1280 m. 5. Press the Start button and click from left to right on suitable points on the front arch of the bridge. If you are of opinion that you have collected sufficient data, press the Stop button. Analysis x What is the shape of the front arch of the bridge (use Analyze > Function Fit). 33 34 II. Modeling Activities Table of Contents 1. INTRODUCTION.................................................................................................................................................... 36 2. INTRODUCTION TO MODELING ............................................................................................................................ 37 Activity: Creating Graphical model - Bathtub .................................................................................................. 37 Activity: Modifying Graphical model – Population growth .............................................................................. 39 Activity: Creating Equations model - Motion of a runner................................................................................. 40 Activity: Introducing Events - Bouncing ball .................................................................................................... 43 3. BIOLOGY MODELS ............................................................................................................................................... 44 Activity: Trees.................................................................................................................................................... 44 Activity: Unrestrained mice growth................................................................................................................... 45 Activity: Carrying capacity ............................................................................................................................... 46 Activity: Weasels eat mice ................................................................................................................................. 47 Activity: Spruce Budworms ............................................................................................................................... 49 Activity: Foxes-Hares model ............................................................................................................................. 51 Activity: Water stream ....................................................................................................................................... 52 4. CHEMISTRY MODELS ........................................................................................................................................... 53 Activity: Model A Æ B....................................................................................................................................... 53 Activity: Model P Æ Q ...................................................................................................................................... 54 Activity: Combustion of carbon monoxide......................................................................................................... 55 Activity: Reaction of magnesium and acid ........................................................................................................ 56 Activity: Crystal violet....................................................................................................................................... 57 5. PHYSICS MODELS................................................................................................................................................. 58 Activity: Bicycle motion..................................................................................................................................... 58 Activity: The fall of the parachute jumper ......................................................................................................... 60 Activity: Damped oscillations............................................................................................................................ 62 Activity: Cooling down a cup of coffee.............................................................................................................. 64 Activity: A lamp connected to the capacitor...................................................................................................... 65 Activity: The rocket............................................................................................................................................ 66 35 1. Introduction Modeling Activities allow using and creating models of dynamical changing systems. Coach offers three modes of constructing and viewing models: Graphical, Equations and Text modes. Graphical mode is based on the stock and flow approach. Models are built with graphical elements: state variables (similar to the stock concept), flows, auxiliary variables and constants. Discrete changes of state variables can be provoked with events. Relationships between model variables are given by connectors and formulas. Dynamic processes of a modeled system are determined by rates of change of state variables6 which in turn are represented mathematically by differential equations. Equations mode is a textual representation of the mathematical equations hidden behind the model structure. These equations consist of differential equations, formulas for calculation of auxiliary variables, set of initial values and constant values. It is possible to create and modify models directly in Equations mode; model equations are constructed by using variable icons. Changes made in Equations mode are automatically made in Graphical mode and vice-versa. The model differential equations of Graphical and Equations mode can be solved by one of the numerical iteration methods: Euler, Runge-Kutta 2 or Runge-Kutta 4. Text mode is also a textual representation of model mathematical equations. In this mode a model is constructed according to the Coach Language rules7; it is actually a program that consists of equations for calculation of the model variables according to the selected iteration method. It is also possible to create and modify models directly In Text mode, a model is built by entering equations which have to be structured according to the rules of Coach Language. This mode is very similar to Text mode of Coach 5. Models created and changed in Text mode can not be viewed in Graphical or Equations mode. The model results can be compared to experimental data (from measurement or video measurement). The model can be easily modified or a parameter can be changed so students can test their hypothesis and make links between the real experiment and the theoretical model. Modeling allows solving realistic problems that cannot be solved analytically at the school level. Models of interesting and complex everyday-life phenomena are possible to create e.g. falling with the air resistance, charging and discharging capacitor, damped and forced oscillations, radioactive decay, etc. The formulation of such numerical models is often rather simple and conceptually easy to understand, and may bear little relation to the mathematical difficulty of getting a solution. For example a simple growth model is rather transparent, while the exponential function, which is the solution, would be considered rather difficult. Similarly oscillator damped by various kinds of frictional force presents considerably mathematical difficulties, while rising only slowly in computational difficulty. There are broadly three possible ways of using Modeling. x Call up the prepared model, run it and discuss with the students how it works. x Start with a prepared model or with parts of a model and build up a new model, after discussion, by revising or adding to what is already there. x Start with nothing, and build up the desired model from scratch, during discussion with the class. 6 7 The rate of change of a state variable is a sum of all in and outflows of a state variable. Coach Language is the programming language of Coach. 36 The first is close to using Modeling as simulation program, with the difference that the model is visible, and can be changed. The Simulation option is very useful in this case. The second is useful in helping students to see how different problems relate to each other. The third is useful for example in modeling a phenomenon being studied in a student investigation. 2. Introductory Modeling Activities Activity: Creating Graphical model - Bathtub In this activity you are going to learn how to construct a simple graphical model. Assume a very simple dynamic system – a bathtub system. The bathtub acts as water container that gains the water through a tap with a faucet and loses it through the drain. Activity procedure x Open Coach 6 Activity Introduction to Modeling > Creating Graphical model – Bathtub. x The Model window is empty. Constructing a model 1. To represent water in the bathtub the state variable is used. x Click the State variable icon in the toolbar of the Model window. The cursor shape changes into the symbol of the state variable. x Move the cursor onto the Model window, position it and click. The variable is labeled State_1. x Rename the variable into Bathtub. The symbol and its name remain selected which allows easy renaming it by overtyping. 2. To represent water flow into and out of the bathtub the flow variables are used. x For the inflow: select the flow icon by clicking on it, position the cursor at the left side of the Bathtub and drag it to the right until it makes contact with the state variable. x For the outflow: select the flow icon by clicking on it, position the cursor inside the Bathtub and drag it to the right. 3. Question marks in the three elements indicate that no numerical quantities or mathematical expressions have been assigned for these variables. Assume that there is 10 liters of water in the bathtub, there is the constant inflow rate of 4 liters per minute and outflow rate of 1.5 liters per minute. x Double click on the Bathtub variable. Its properties window opens. Enter unit (l) and Initial value (10). If you like you can give an extra description of this variable. This description together with the initial value will be displayed when mouse cursor is placed over the Bathtub graphical symbol. Click OK. x Double click on the inflow. Name it Fill and check the option Show name. Enter unit (l/min), and definition (=4). Click OK. x Double click on the outflow. Name it Drain and check the option Show name. Enter unit (l/min), and definition (=1.5). Click OK. 4. By default time t with unit s is used as the independent variable. You are interested to see the bathtub volume change in minutes. 37 5. 6. 7. 8. x Click the Independent variable icon and change the unit from s into min. Click OK. To see how the bathtub volume responds to the inflow and the outflow right click the Bathtub symbol and select the Display as diagram option. Click the left, upper pane to place the diagram. In the same way display the diagrams of Fill and Drain flows. Use other two empty panes. The length of the run can be changed in the Model settings window. Click the Settings button available in the Activity toolbar. Change the Stop time into 60 with a time step of 0.25. This step interval is the increment of time over which the program does the calculations specified by the model. You may notice that in the Model Settings window you can also set the integration method that the program uses to make the calculations. Euler's method is usually the default method. The Runge-Kutta methods give more accurate results. It is often a good idea to run your models with a variety of step intervals and integration methods to be sure that your results are not an artifact of the calculation method. Executing a model 1. First think about how this system might evolve – try to predict how the Bathtub volume, the Fill flow and the Drain flow change over time. 2. Use the Sketch option to draw your predictions. 3. Execute your model by clicking the Start button. The computer will plot the graphs. Use the Zoom to fit button to adjust the scale of your diagram. 4. Right click the Model window and check the Run controller option. You can use the time slider to step through the time values. 5. What kind of function describes the Bathtub volume versus time graph? You can use the Function Fit option available in the Tool menu of the Diagram pane. Can you conclude what is the meaning of the slope of the bathtub graph? 6. Investigate the graphs for different Fill and Drain rates. x What happens to the water level in the bathtub when the Fill is set zero? x What happens to the water level in the bathtub when the Drain is set to zero? x What component of an actual bathtub is missing that prevents disaster in previous case? 7. Now assume that the Fill flow changes in time, the first 20 minutes the flow rate is 5 liter per minute and then it becomes 2 liter per minute. Change your model by using condition for the Fill flow. Describe the resulting graphs. Modifying the model Modifying a model to better simulate a real system is an important part of the scientific process. Let's try to improve the bathtub model. Actually the rate of flow through drain is not constant. For a given drain opening, the more water in the tub, the grater the rate of flow. The rate of flow is a function of water volume in the tub. 1. You are going to change your model to reflect the relationship between the drain rate and bathtub volume. Such dependence can be shown in the model by using connectors. x Select the Connector icon. x Place the cursor inside the Bathtub symbol. x Click and hold the mouse button. Drag the cursor out of the state variable until it makes contact with the Drain flow. Release you click. x If you are picky about the arc traced by your connector, you can bend it in any way you like. To do so click the connector and use 4 given points; 2 points which stick to 38 2. 3. 4. 5. 6. the variable allow sliding around the variable edges. The other 2 points allow bending the connector. Redefine your Fill and Drain flows. First assume that the faucet is closed. x Double click the Fill and set it to 0 liters per minute. x Double click on the Drain. Replace the constant value with the equation Drain = 0.1 * Bathtub. Use the Formula Editor to construct your formula. Use the Sketch option to predict the results of your model. Execute your model. What kind of function describes the Bathtub volume graph? You can use the Function Fit option available via the Tool menu of the Diagram pane. How would you change your Fill flow rates to keep the same amount of water in bathtub? Activity: Modifying Graphical model – Population growth In this activity you are going to investigate the population growth. This example is used to teach you how to modify graphical models. Activity procedure x Open Coach 6 Activity Introduction to Modeling > Modifying Graphical model – Population growth. x The Model window displays a simple model of the population growth in a small Dutch town. The town has 5000 inhabitants. Every year there are 150 babies born in the town and 75 people die. x Execute the model by clicking the Start button. x Describe how the population is changing in time. This model is not very realistic; the assumption was made that the birth and death rates in the town are constant. In real life both the birth and death rates of a population depends on the current size of that population. Let's modify the model. Modifying the model 1. Birth rate is a fraction of the existing population. It represents the fertility of the population. The birth fraction is calculated by dividing the average birth rate by the average population. In our case it would be 0.03 per year (3% per year). You can put the birth fraction into your model by using the constant variable. x Select the Constant. Position the variable symbol below and the left of Birth flow. Click to place it. Label it Birth_fraction. 2. You know that the Births flow depends on the birth fraction, so you need a connector linking the Birth_fraction to Births. x Select the Connector icon. Click the Birth_fraction symbol and drag the connector until it makes contact with the Births flow. Release you click. 3. The Births depends also on the total population, so you need a connector linking the Population to Births. x Select the Connector icon. Click the Population symbol and drag the connector until it makes contact with the Births flow. Release you click. 4. Notice that now Births contains a ? again. The previous equation is not valid anymore. Connectors drawn from Population and Birth_fraction to Births inform us that the equation for Births must contain the variable population and Birth fraction constant. 39 x 5. 6. 7. 8. 9. Double click the Births flow and redefine it. Use the Formula Editor. Notice that only connected variables are available in the Formula Editor. Make similar changes to the Deaths flow. The Death_fraction represents the mortality of the population. The calculated Death_fraction for our population is 0.015 or (1.5%). Run you model. For the execution of the model you can also use the Run controller. Describe the way the population growths. Use the Function Fit option to find the function which fits your graph. Use the Simulate option available in the Model window menu to simulate different values of births and death fractions. x What happens to the total population in a situation where births fraction is greater than the death fraction? x What happens to the total population in a situation where births fraction is lower than the death fraction? There are two feedback loops in this model; can you tell which variables are involved in these loops? Activity: Creating Equations model - Motion of a runner In this activity you will learn how to create a dynamical model in Equations mode. For this you will use the model of a runner. In Equations mode, instead of graphical symbols the mathematical equations are used. These equations describe the distance covered by the runner and its velocity over time. It is not possible to type equations directly; they are constructed by using the Variable buttons available in the toolbar of the Model window. Activity procedure x Open Coach 6 Activity Introduction to Modeling > Creating Equations model – Motion of a runner. x The Model window is empty. Constructing the model 1. To represent a Distance traveled by the runner the state variable is used. x Click the State variable icon in the toolbar of the Model window. Its Properties dialog opens. x Fill the variable properties: name Distance, unit m, Initial value 0. x The Distance is growing with a certain rate of change (Inflow). Click the Add inflow button and name it Rate_Of_Change_Of_Distance (for explanation see also Model explanation below). As a result the State and Flow variable equations are placed in the left pane of the Model window and the Initial value equation is placed in the right pane of the window. Notice that the Flow is not defined yet, it displays ‘?‘ sign. You will define it later. 2. From your physics lessons you know that the rate of change of distance of a moving object is actually its velocity. x Double click the Flow equation and fill as definition Velocity. 3. Velocity variable is not defined in your model yet, Coach can not “recognize it” and displays the ‘?’ sign at the beginning of the equation. Let's assume that the runner runs with the constant velocity of 5 m/s. 40 4. 5. 6. 7. x Double click the Velocity equation. x Select the Constant variable type and click OK. x The Constant variable properties window open. Fill in unit m/s and the constant value 5. Notice that the Velocity equation was moved to the Initial values pane. To see how the distance changes in time right click the Distance equation and select the Display as diagram option. x Click the right upper pane to place the diagram. In the same way display the diagram of Velocity versus time. Use other empty pane. The length of the run can be changed in the Model settings dialog. Press the Settings button available on the Activity toolbar and change the stop time into 20 with a time step of 0.1. This step interval is the increment of time over which the program does the calculations specified by the model. You may notice that in the Model settings window you can set the integration method that the program uses to make the calculations. In this example the Euler's method is used. Executing the model 1. First think about how this system might evolve in time - try to predict how the Distance and Velocity change in time. 2. Use the Sketch option to draw your predictions. 3. Execute your model by clicking the Start button. The computer will plot the graphs. Use the Zoom to fit button to adjust the scale of your diagram. 4. Right click the Model window and check the Run Controller option. You can use the time slider to step through the time values. 5. What kind of function describes the Distance versus time graph? You can use the Function Fit option available in the Diagram/Table pane. 6. Use the Simulate option available in the Tool menu of the Model window to simulate different velocity values. 7. What is the meaning of the slope and y-intercept of the Distance versus time graph? Modifying the model Let's assume that the velocity of the runner is not constant, the runner accelerates at the starting phase of his motion. You are going to change the model to reflect that the runner velocity is not constant. 1. Remove the Velocity constant from your model. x Select the Velocity equation in the Initial values pane and delete it with <Del> key. 2. To represent the velocity which is changing in time the State variable has to be used. x Click the State variable icon. Its properties window opens. x Fill the variable properties: name Velocity, unit m/s, initial value 0. x The Velocity is growing with the certain rate of change (Inflow). Click the Add inflow button and name it Rate_Of_Change_Of_Velocity. 3. From physics you know that the acceleration of a moving object is the rate of change of its velocity. x Double click the Flow equation and fill as definition Acceleration. 4. Acceleration variable is not defined yet, Coach displays the ‘?’ sign at the beginning of the equation. Assume that the runner accelerates with the constant acceleration of 1.5 m/s2. x Double click the Acceleration equation. x Select the Constant variable type and click OK. 41 5. 6. 7. 8. 9. x The Constant variable properties window opens. Fill in unit m/s2 and the constant value 1.5. Use the Predict option to predict the results of your model. Execute your new model. What kind of function describes the Distance versus time graph? You can use the Function Fit option available in the Diagram pane. Use the Simulate option available in the Model window menu to simulate different Acceleration values. What is the meaning of the slope and y-intercept of the Velocity versus time graph? Model explanation The model describes the motion of a runner. Time increments by a small interval dt t = t + dt this equation is hidden and not shown in the Model equations. Distance traveled by the runner is calculated based on the formula: Distance(t) = Distance(t-dt) + dDistance which means that the new Distance is the previous Distance plus small distance dDistance covered by the runner in the small time interval dt. For the first calculations the initial Distance value 0 m is used. The runner runs with a certain Velocity which can be calculated as the distance dDistance traveled in the time interval dt Velocity= dDistance /dt, in other words the Velocity is the rate of change of Distance. Then Distance can be described as Distance(t) = Distance (t-dt) + Velocity*dt In the language of Modeling the state variable is defined by its rate of change: State(t) = State(t-dt) + (dState/dt) * dt where the rate of change dState/dt (the change of the State in the time interval dt) is the total sum of all inflows and outflows. For the Distance variable we can assume that its rate of change is determined only by one inflow (positive change in state variable) which is equal to the Velocity. When the velocity of the runner is changing in time then the rate of change of the velocity is its acceleration. 42 Activity: Introducing Events - Bouncing ball In this activity you are going to learn when you can use Events and how to construct them. You are going to create a simple model of a bouncing ball. Assume you are dropping a ball from a height h onto a hard floor. When it bounces off it loses energy, its speed is reduced significantly by its interaction with the floor. The ball is characterized by its coefficient of restitution (the ratio of its rebound speed to its collision speed) which in this model has a value of 0.9. Activity procedure x Open Coach 6 Activity Introduction to Modeling > Introducing Events – Bouncing ball. x In the Model window you see the model of free falling body. This model describes uniform accelerated motion with acceleration g. The diagrams of height and velocity of the falling ball are prepared. Execute the model to see the graphs. x You are going to modify this model into a model of a bouncing ball. Modifying the model into bouncing ball model At the moment of bouncing the ball velocity changes its direction and speed in the following way: rebound speed = coefficient of restitution * collision speed. In the model of uniform accelerated motion the ball velocity is determined by its acceleration (the rate of change of velocity). At the moment of bouncing the velocity can not be calculated from this equation, the velocity changes its direction and value. It is rather a discrete change of velocity based on a certain condition. Such change can be given with Event. When the ball reaches the ground (trigger condition: height =< 0) then velocity changes its direction and value (velocity= - 0.9 * velocity). 1. Click the Event icon on the toolbar of the Model window, position the cursor at the right of the existing model and click. The symbol of Event is placed in the Model window. 2. Double-click the Event symbol and define its properties. Event definition is built from two parts, trigger condition and list of actions. x Define the trigger condition by entering the condition height <= 0 equation in the On field. The relation can by typed in or created with the Formula Editor. x Define the action by clicking the Add button, selecting the velocity variable from the list and entering its definition (=-0.9*velocity). x Click OK to close the Properties dialog. 3. As a result the connectors to variables used in Event are created. 4. In this model you are going to use the stop condition. Click the Settings button and select condition. The condition is already predefined, explain what it means. 5. Execute the model. 6. Click the Show in Equations mode button to see model equations. Notice that Event makes use of Coach Language conditional statement On condition Do statements EndDo. 43 3. Biology models Activity: Trees In the forest are 4000 trees. The forest administrator would like to earn some money by cutting old and planting new trees. He cuts 20% of the trees per year and plant every year 1000 new trees. Activity procedure 1. Open Coach 6 Activity Biology models > Trees. 2. Analyze the model and perform assignments. Model explanation The number of trees in the forest is given by the State variable Trees. The change in the number of Trees is difference between the number of planted trees and the number of cut trees. Since every year 20% of all trees disappear, the CuttingFactor is 0.2. The constant amount of 1000 trees is planted each year. Assignments 1. Look at the model and predict the graph. x Use the Sketch option available in the Tool menu of the Diagram pane. x Start the model. Is your prediction correct? 2. After how many years the number of trees stabilizes? Use the Scan option available in the hammer menu of the Diagram pane. 3. Calculate how many percent of the trees may be cut to keep the constant number of 4000 trees. 44 Activity: Unrestrained mice growth Mice have a short reproduction period. This means, that if you start with a small mice population it will grow fast. The mice in the population will get offspring, this offspring will mature and these matured young will breed themselves. The population will grow exponentially. Activity procedure 1. Open Coach 6 Activity Biology models > Unrestrained mice growth. 2. Execute the model by pressing the green Start button. 3. Analyze the model and perform assignments. Model explanation The change of mice population is determined by the number of new born mice and the number of death mice. The initial mice population is 10. Mice population changes in time in the following way: Mice(t) = Mice(t-dt) + (Births-Deaths)*dt The amount of births and deaths depend on the current size of that mice population. Births = Growth_factor*Mice Deaths = Death_factor*Mice In one period from 100 mice 100 mature offspring originates, the growth factor is 1. In one period from 100 mice 0 mice will die, the death factor is 0. Assignments 1. Determine after how many years does the amount of mice exceed acceptable level (amount > 350)? Determine after how many years the amount of mice exceeds half a million (use the Zoom to fit button). 2. Before giving birth to a new offspring, 30 of every 100 mice will die. Adjust the initial values to this fact. Does the amount of mice still exceed a certain acceptable growth level (amount > 350)? Find out after which period a population of 500 mice disappears if they do not give births anymore. 3. Is it possible, with the right combination of variables for the birth- and death rate, to have a stable population? What is your conclusion? 45 Activity: Carrying capacity This activity is the continuation of the Unrestrained mice growth activity. In a mice population the growth rate depends on the carrying capacity of the environment. The carrying capacity is the maximum amount of mice the garden can accommodate. In this model you will see how the amount of mice grows but will never exceed the carrying capacity. Ten mice live in a garden (M=10). Nevertheless there is a limited amount of food. The maximum number of mice the garden can accommodate is called the carrying capacity. Is the size of the garden an effective restraining for mice growth? Activity procedure 1. Open Coach 6 Activity Biology models > Carrying capacity. 2. Execute the model by pressing the Start button. 3. Analyze the model and perform assignments. Model explanation Growth factor causes the increase in amount of mature mice if no restrictions are imposed. In one period from 100 mice 100 mature offspring originates, the growth factor is 1. The capacity is the optimal amount of mice that can live in the garden. The mortality is high when the capacity is small compared to the amount Mice. Then the amount of Mice divided by Capacity is >> 1. The mortality is small if the Capacity is high compared to the amount Mice. Then the amount of Mice divided by Capacity is << 1. The relation Mice/Capacity is a measure for the Restraining. Assignments 1. How the graph shows that restrained growth take place? Find out what is the effect of doubling or halving the initial values. What is your conclusion? 2. The size of the garden restrains the population growth. Which variable in the model shows this? x Explain how the restrained growth is implemented in the model. x Explain why no death rate is implemented in the model. x Predict at which value the population will stabilize if there is 80 young born mice per 100 mice and garden capacity is 75 mice. 3. Is this model suitable for describing the development of the mice population? Explain your answer. 46 Activity: Weasels eat mice This activity is the continuation of the Carrying capacity activity. Many animals have natural enemies; weasels eat mice. If there are a lot of mice, there is a lot of food for weasels. If there are a lot of weasels, a lot of mice are eaten. This affects the number of mice. The decrease in the food for weasels affects the number of weasels. With a few weasels, mice population grows again, and so on. Assume that ten mice live in a garden (M=10) and the carrying capacity of the garden is 100, are the weasels able to stabilize the number of mice? Activity procedure 1. Open Coach 6 Activity Biology models > Weasels eat mice. 2. Execute the model by pressing the green Start button. 3. Analyze the model and perform assignments. Model explanation Restrained growth of mice The mice population depends on three factors: x the mice reproduction, x the capacity of the garden, x the presence of a predator. Unrestrained growth of the mice is given by growth factor G = 1 (100 mice are born per 100 mice) Restrained growth of the mice is given by restraining R = M/V where capacity V is 100. The mice population depends also on the amount of weasels and on their success. There are 20 lucky weasels per 100 meetings (O=0.2). Growth of weasels The growth of the weasels is given by growth factor A = 0.03 (3 weasels are born per 100 weasels). The population of weasels will increase if there are more mice. Death rate of the weasels is not influenced by the number of mice only given by the death factor D = 0.6 (60 deaths per 100). Assignments 1. At which moment does the weasels population grows fastest compared to the number of mice? At which moment is the decrease in weasel number highest compared to the number of mice? 47 x 2. 3. 4. 5. Compare the equilibrium at which the number of mice stabilizes with the results from the other models (2 previous Activities). Increase the carrying capacity of the garden 2 times. Calculate with the model at which equilibrium the number of mice and weasels stabilizes. Increase the success rate at which the weasels catch mice 2 times. x Calculate with the model at which equilibrium the number of mice and weasels stabilizes. x These results and the results of assignment 2 are remarkable. Explain why? In this model, the mice population is completely kept under control by the presence of weasels. Even if weasels hardly catch mice and if the carrying capacity of the garden is high. x Show that in this model, after a long period, the number of mice only depends on the growth and the death rate of the weasels. x The amount of weasels is much smaller than the amount of mice. Is this realistic? Explain why this model is well/not useful to describe the growth of the mice and weasels population. 48 Activity: Spruce Budworms The spruce Budworm (Choristoneura fumiferana) lives in northern coniferous forests and its number irrupts at approximately every forty years. If the population grows it causes much damage for the wood industry. Budworms eat leaves. Their density depends on the carrying capacity of the forests (which depends on the average leaf area per tree). The model in this activity describes the change of the Budworm population in time. The changes in the population are determined by the number of new born Budworms and the number of dead Budworms. The deaths of Budworms are caused mainly by birds. In the model is considered the predator pressure as constant. Activity procedure 1. Open Coach 6 Activity Biology models > Spruce Budworms. 2. Execute the model by pressing the Start button. 3. Analyze the model and perform assignments. Model explanation For the growth of the Budworm population the following rules are taken into account: At the low density of Budworms the number of births increases together with the size of the population. At high density the population size is limited and has its maximum size. The following formula describes the population growth = r*N*(1-N/K) where N = is the number of Budworms r = growth rate of the Budworms K = carrying capacity of the forest. The Budworms are killed mostly by birds. The number of birds in the forest is not determined by the number of Budworms because there are also alternative food sources. The number of birds remains the same. When the density of Budworms is low then birds have difficulties to find them and eat less of them. Birds eat more worms when the density of Budworms is high. Birds have maximal amount of Budworms which they can consume. Since the number of birds remains the same then the number of eaten Budworms reaches a maximum. The predation term is given by E*P*N^2/(O^2 + N^2) where E = maximal efficiency of the predator P = predation pressure O = Budworms density at which the predator pressure (P) is optimal 49 K and O are both composed of a constant (respectively c and f) and the average leaf area per tree (A). K=c*A and O=f*A Assignments x Using an initial population of 30 spruce Budworms, determine the stable equilibrium for the Budworm population for the following values of A = 1700, 2150 and 2500. x Repeat the analysis done above for the same values of A but with an initial population of 25000 budworms. When comparing your results with those from the previous problem, you should notice something interesting. x Use A = 2150. Try to determine between which (whole) values of N the graph "jumps" to the other equilibrium. x Explain the changes in height at which the equilibrium stabilizes, when N is constant and A varies. What does this mean in the biological situation? 50 Activity: Foxes-Hares model One of nature's rules is eat, or be eaten. It is not unusual that one species lives by eating another species. When looking at a population of arctic hares and a population of arctic foxes, it will be clear that the hares serve as food for the foxes, in other words, the fox is the predator, and the hare is the prey. In this activity you are going to use predator-prey model. Activity procedure 1. Open Coach 6 Activity Biology models > Foxes-Hares model. 2. Execute the model by pressing the Start button. 3. Analyze the model and perform assignments. Model explanation The model consists of two populations, the Hares population and Foxes population. M is the growth factor of the hare population. K is the death factor of the hare population. Foxes kill hares so the total number of hare deaths depends also on the number foxes. G is the growth factor of the fox population. Foxes need hares to survive, so the total number of fox births depends also on the number of hares. D is the death factor of the fox population. Assignments 1. How large is the population of hares and foxes after 15 months? x How large is the population of hares and fixes after 22 and 31 months? 2. Explain the graph of the number of foxes versus the number of hares. 51 Activity: Water stream In and around a water stream live many different organisms. Both plants and animals have their own natural habitat. Think, for example, of reed, algae, water thyme, fish, birds and insects. All these plants and animals live together and provide each other with nutrients. The plants produce oxygen which is of vital importance for the animals. At night, the plants themselves also require oxygen. The dissolved oxygen concentration in the water, which increases strongly during the day, decreases during the night. The model in this activity calculates the oxygen concentration in the water over time. You can study furthermore which factors influence the oxygen levels in the water. Activity procedure 1. Open Coach 6 Activity Biology models > Water stream. 2. Execute the model by pressing the Start button. 3. Analyze the model and perform assignments. Model explanation In the model a number of factors of the natural habitat are calculated. Some of these depend on each other. In photosynthesis plants produce oxygen. The light intensity influences photosynthesis, the higher the light intensity, the higher the photosynthesis rate. Oxygen in water is being used in respiration process as well by plants and as by animals. The respiration is influenced by the oxygen concentration and the number of plants and animals. The light is defined such that there is a day and night rhythm. Assignments 1. You can explore the influence of a specific parameter in the model by right clicking the Model window and selecting the Simulate option. Select a new variable value and click Start. The graph corresponding to the entered value is drawn. x Explore the influence of the variable Animals. Explain the results. x Explore the influence of the variable Plants. Explain the results. 52 4. Chemistry models Activity: Model A Æ B A small but very harmful amount of compound A pollutes a pond. It reacts with water to form the harmless compound B: A + H2O Æ B Compound A does not disappear immediately. How do A and B change in time? Activity procedure 1. Open Coach 6 Activity Chemistry models > Model A Æ B. 2. Analyze the model and perform assignments. Model Explanation Decrease of [A] is equal to increase of [B]. The reaction rate v is proportional to [A]. v = k*[A] Water is in excess so [H2O] is about constant. Even if the reaction rate is also proportional to [H2O], so v=k'*[H2O]*[A] then still v=k*[A] and k'*[H2O] = k. Assignments 1. The diagram shows you the decrease of [A]. Predict the change of [B]. x Right click the Diagram pane and select Sketch. Draw the change of [B]. Press <Esc> to finish your prediction. 2. Display a graph of the change of [B]. x Right click the Diagram pane and select the Create/edit diagram option. Click on C3 and choose Axis, Second vertical. Click OK. x Does the curve agree with your prediction? Note for this reaction: Decrease of [A] = Increase of [B] x Explain how it is given in your model. x What relationship has been assumed between [A] and the reaction rate (v). x Why does [H2O] not appear in the rate equation? 3. The reaction will be faster at a higher temperature. x Change an initial value to simulate a reaction at a higher temperature. x How do the curves show you that the reaction takes place faster? x Simulate also a reaction at a lower temperature. 4. (Use the original initial values). After some time [A] will be half the initial value. This time is half-live time of [A] for this reaction. x Determine in the diagram the half-live time of [A]. x Study the graph to examine if the half-live time of [A] also depends on [A]. What is your conclusion? 5. The safety level of compound A is [A] < 4 mmol/l. x Estimate using the half-live time at what time the safety level has been reached. Check your answer using the graph. 53 6. In the model proportionality has been assumed between reaction rate and concentration. x Change this assumption in the model into: the reaction rate is proportional to the square of [A] ([A]2). Examine again the relation between concentration and the half-live time of [A]. What is your conclusion? Activity: Model P Æ Q A compound P reacts to a compound Q: P Æ Q. Does this chemical equation also describe the process during the reaction? Activity procedure 1. Open Coach 6 Activity Chemistry models > Model P Æ Q. 2. Analyze the model and perform assignments. Model Explanation The model has been composed from two parts. Each part is identical to the model from the previous Activity: Model A Æ B. Assignments 1. Click the Start button to execute the model and show the change of [P] and [Q] in the diagram. The mixture of P and Q has a volume of 500 ml. Check in the diagram: ... mol of compound P at t=400 s has been converted into ... mol of compound Q. 2. Atoms disappeared? Before you continue, think about what could have happened with compound P. Continue after you have formulated a suggestion. 3. Compounds disappear but atoms normally don't, so compound P cannot directly form a compound Q. It is necessary to assume an intermediary compound I: Step 1: P Æ I Step 2: I Æ Q x Predict the change of [I] (Use the Sketch option in the Diagram menu). x Show [I] in the diagram by selecting Create/Edit diagram. Choose C3 and set Axis to First vertical. x Does the [I]-curve agree with your prediction? 4. Study the model. Note the distinct rate constant for every reaction step. Simulate graphs with kQ = 0.5. The graph looks like the reaction A Æ B from Activity: Model A Æ B. Suggest reasons for this similarity. 5. Simulate graphs with kQ = 0.0000000005 (or 5e-10). x Again the graph looks like the reaction A Æ B from Activity: Model A Æ B. Suggest reasons for this similarity. x Someone proposes that this situation actually describes the reaction P Æ I? Is this right in an every day practical situation? 6. The step, prescribing the rate of formation of compound Q is the rate that determines step. x What is the rate determining step in problem 4? Which step is it in problem 5? 54 Activity: Combustion of carbon monoxide Exhaust gas still contains CO. To convert CO into CO2 the gas must be heated with sufficient air (oxygen in excess). During a day it is allowed to be exposed to a CO concentration of 50 ppm (55mg/m3). Is it possible to remove sufficient CO from exhaust gas to be safe for daily exposure e.g. exhausting gas from an indoor lift-truck? Activity procedure 1. Open Coach 6 Activity Chemistry models > Combustion of carbon monoxide. 2. Analyze the model and perform assignments. Model Explanation Reaction rate v is proportional to as well to [CO] as to and [O2]. v = k*[CO]*[O2] A decrease of [CO] = an increase of [CO2] d[CO] = -d[CO2] A decrease of [O2] = an increase of [CO2]/2 d[O2] = -d[CO2]/2 Notice in the model how the rate constant k depends from temperature. Assignments 1. A model describes the conversion of CO in CO2 at 1125 K. x Click on the green Start button to execute the model. x Determine from the graph how long the mixture must stay at 1125 K to reach the concentration for daily allowed exposure. 2. To treat as much exhaust gas as possible it has to stay in the reactor as short as possible. Show by making graphs. x 10 % increase of [O2] is hardly effective. x 50 % decrease of [CO] has no sense. x 10 % increase of the temperature is very effective. 3. To treat as much as possible exhaust gas it must stay at most 10 s in the reactor. x Determine with an accuracy of 5 K what the minimum temperature in the reactor should be. x Use the original initial values for the concentrations: [O2]= 5 mol/m3 [CO] = 0.04 mol/m3 [CO2]= 0.00 mol/m3. 55 Activity: Reaction of magnesium and acid The model in this activity simulates the decrease of [H+] during the reaction of Mg and H+. The model contains an assumption about the reaction rate (v). You must verify this assumption according to measurement results. Activity procedure 1. Open Coach 6 Activity Chemistry models > Reaction of magnesium and acid. 2. Analyze the model and perform assignments. Model Explanation During the reaction of Mg and H+ the number of [H+] decreases. In the model it is assumed that that the reaction rate v is proportional to [H+]. v = k*[H+] Assignments 1. Which variable determines the change of [H+]? Formulate the assumption you must check by comparing with measurement results. 2. Execute the model. 3. Import the data from Result Reaction of magnesium and sulfuric acid. Use the Import Background Graph… option available via the Tool menu of the Diagram pane. x Compare the calculated graph [H+] and the measured graph? x Which initial value are you allowed to change in the model to fit the calculated and the measured graph? x Use the Simulate option, available via the Tool menu of the Modeling window, to create graphs for different values of a variable. x What is your conclusion about the relationship between the rate of reaction and concentration of [H+] during the reaction? 4. Use the model to check if another assumption also results in fitting curves. x Check for instance the assumption v = k*[H+]*[H+]. Redefine the formula of the reaction rate by double clicking the flow v and typing the new formula. x What is your conclusion? 5. Redefine the formula of the reaction rate. Double click the flow v and type a new formula. x Check with the model how -log([H+]) changes. x Check also for the measured data how -log([H+]) changes. x What are your conclusions? 56 Activity: Crystal violet The model used in this activity simulates the change of [KV] during the reaction between KV and OH-. Activity procedure 1. Open Coach 6 Activity Chemistry models > Crystal violet. 2. Analyze the model and perform assignments. Model Explanation During the reaction KV and OH- the number of [KV] decreases in time. The assumption used in the model is that the reaction rate v is proportional to [KV]. v = k*[KV] Assignments 1. Which assumption does the model contain about the rate of change of [KV]? 2. Execute the model. 3. Import the data from Result Crystal violet. Use the Import Background Graph option available via the Tool menu of the Diagram pane. x Does the calculated curve of [KV] looks like the measured one? x Which initial value are you allowed to change in the model to enhance the fit? x Use the Simulate option, available via the Tool menu of the Modeling window, to find a value for the best fitting curve. x Should the whole graph agree with the measurement results? (What do you know about the colorimeter?) x Determine the moments between which both curves should fit. x What is your conclusion about the relationship between [KV] and the reaction rate for the reaction between [KV] and [OH-]? x What is your conclusion about the relationship between [OH-] and the reaction rate? 4. Create and describe the curve of -log([KV]) for the model results and for the measurement results. 57 5. Physics models Activity: Bicycle motion Cycling; you probably know all about it! You know that when you exert forces on the pedals, you provide the required forward force. On the basis of what you have learned about force and acceleration, you might expect that you will go faster and faster when you exert such forces. But in reality you don't. That's because there are other forces (air resistance, rolling resistance) that prevent the speed of your bicycle from continuing to increase. In this activity you are going to make a model that describes the motion of a bicycle. Activity procedure 1. Open Coach 6 Activity Physics models > Bicycle motion. 2. Analyze the model and perform assignments. Model Explanation The prepared model describes the accelerated motion. The position of the bicyclist x is increasing in time. The change of position dx in a small time interval dt is defined by its rate of change – velocity v. x(t) = x(t-dt) + (dx/dt)*dt and dx/dt = v The velocity of the bicycle v is increasing in time. The change of velocity dv in a small time interval dt is defined by its rate of change – acceleration a. v(t) = v(t-dt) + (dv/dt)*dt and dv/dt = a The acceleration a is constant. Assignments Assume that the mass of a bicyclist and his bicycle is 75 kg. Bicyclist starts from rest, and exerts an average force of 100 N on one or the other pedal. The net force acting on a bicyclist and his bicycle can be described as: F=Fped/6 - m*g/150 - 0.5*v2 To better understand this equation read Forces on a bicycle during its motion. The net force F on the bicycle leads to the acceleration a of the bicycle a = F/m. 1. Modify your model by adding the new variables, creating the relations between them and defining them. 2. If the bicyclist keeps cycling with a constant (average) pedaling force of 100 N, what is his final speed? 3. Which average force should he exert on his pedals in order to reach a final speed of 8.0 m/s? To solve this problem, you have to try out different values of Fped, until you get the one that 58 gives you the desired value of the final speed. Use the Simulate option available via the Tool menu of the Modeling window. 4. What average pedaling force is needed to reach a final speed of 8.0 m/s? Forces on a bicycle during its motion As soon as the bicyclist has some speed, the air resistance (Fair) and the rolling resistance (Froll) are going to play their part. They oppose the forward force produced by the bicyclist (Fforward), and slow his motion down. The forward force is caused by the pedaling force (Fped) exerted by the bicyclist on the pedals of his bicycle. Through the chain, this force causes the wheel to move and the wheel then exerts a backward force on the road surface. But action = - reaction, so the road surface exerts an equally large forward force on the bicycle. The rotation of the pedals is transferred to the wheel, which starts the bicycle moving. The bicycle moves much faster than the pedals, but this means that the force on the pedals (averaged over one revolution) is much larger than the forward force on the bicycle. That ratio depends on the transmission rate; values of 4 to 8 are common. So it seems reasonable to choose a value of 6. Fforward = Fped /6 The rolling resistance turns out to be independent of the speed of the bicycle, but it does depend on the weight G = m*g (m = mass in kg, g = 9.81 m/s2) of the bicyclist + bicycle. A reasonable (average) value is given by Froll = m*g /150 The air resistance, however, strongly depends on the bicycle's speed, and hardly at all on its mass. The air resistance is proportional to the square of the speed. The value of the proportionality constant depends on whether the cyclist sits upright or bends over, and varies between 0.2 and 1.0. As an average, take 0.5: Fair=0.5*v2 The net force F on the bicycle, caused by all these forces together, leads to the acceleration a of the bicycle: a=F/m 59 Activity: The fall of the parachute jumper When a parachute jumper comes down, at first he moves downwards faster and faster. But fortunately, his parachute slows him down, and after a while he reaches a constant speed. That is also the speed with which he reaches the ground. In this activity you are going to investigate how that terminal speed of the parachute jumper depends on his weight and on the air resistance on his parachute. Activity procedure 1. Open Coach 6 Activity Physics models > The fall of the parachute jumper. 2. Analyze the model and perform assignments. Model Explanation When the parachute jumper jumps, his plane is flying at a certain altitude h. His vertical speed (downwards) at time t will be called v. The vertical distance he has traveled at time t will be called x; so at time t = 0 he starts off with x = 0, and x never exceeds h. As soon as he has jumped, his parachute unfolds. The amount of time needed for this will be neglected. To meet the regulations required by law, his parachute must have unfolded completely by the time he reaches a height of 700 meters above the ground. This means the plane cannot possibly have flown at an altitude of less than 700 meters. The mass of the parachute jumper is called m (that includes the mass of his parachute, which is usually no more than 2 kg). He is pulled downward by gravity (his weight), but slowed down by the air resistance on himself and his parachute. The net force acting on the parachute jumper is F=Fg - Fr That force therefore depends on m and on the diameter d of the parachute (since d determines the air resistance) F=g*m - 6d2v2 The acceleration of the parachute jumper is a=F/m. Assignments 1. Describe how the speed of the parachute jumper changes during his fall. x Describe how the net force on the parachute jumper changes during his fall. 60 x What can you say about the speed of the parachute jumper when the net force F has become equal to zero? x Once the parachute jumper has reached his terminal speed, what relation is there between the force of gravity acting on him, and the air resistance? 2. The graphs have been calculated for certain initial values of the quantities involved. Find those. x What is the mass of the parachute jumper in the graphs shown? x And the diameter of his parachute? (Remember that all values given were to be expressed in S.I. units!). 3. You are going to find out what terminal speeds are reached by parachute jumpers with different mass. Assume they all use the same parachute, so the value of the diameter d is the same for all. x Use the Simulate option and change the value of m, while d remains at 2.5 meters. x For each of the m-values (10; 20; 50; 70; 90 and 110 kg) read the parachute jumper's terminal speed vend by using the Scan option. Calculate v2end from each measured value, and enter it in the Notes window. x Make two graphs in one diagram: one graph of vend vs. m, and one graph of v2end vs. m. What do you conclude about the relationship between Vend and m? 4. You are going to determine the terminal speeds reached by parachutes of different diameters d. Assume that, in each case, it is you who is hanging from the parachute, so that the mass m always has the same value. For that value, you may choose your own mass (in kg) plus 2 kg for the mass of the parachute. x Use the Simulate option and proceed in the same way as in the previous assignment, but with d as variable instead of m. Carry out measurements for the following dvalues (3.5; 3.0; 2.5; 2.0; 1.5; 1.0; 0.5; 0.20 m). x Start at d = 3.5 m, and proceed to smaller d-values. When you are dealing with the lowest d-value (0.20 m), you have to move downwards by several hundred meters before reaching your terminal speed! Of course, a parachute as small as that doesn't exist; this is (approximately) the situation when you fall down without any parachute, and it is only the air resistance on your body that slows you down (hopefully, this will never happen to you). x Make two graphs in one diagram: one graph of vend vs. d, and one graph of vend vs. 1/d. What do you conclude about the relationship between vend and d? 61 Activity: Damped oscillations An ideal mass-spring system would oscillate indefinitely. But in the real world friction retards the motion of vibrating mass and the mass-spring system comes to rest. This effect is called damping. In this activity you are going to modify model of harmonic motion and construct a model of damped motion which will describe the real experiment. Activity procedure 1. Open Coach 6 Activity Physics models > Damped oscillations. 2. Analyze the model and perform assignments. Model Explanation The model in the Model window is the model of harmonic motion. From the second Newton's law the mass is moving with acceleration a=F/m. When the spring is stretched away from its equilibrium position the spring force is proportional to the vertical displacement x. F= -kx where k is a spring constant The negative sign signifies that the direction of spring force is always opposite the direction of the mass's displacement. To see model equations press Show in Equations mode button. Assignments 1. Click the Start button to execute the model. Describe the graph. x Load as the background graph the experimental data form Damped oscillation Result. The experiment is described below. x Compare the graphs. Explain differences. 2. Which friction force should be taken into account in the described experiment? Text Damping can help you to answer this question. x Form a hypothesis that this friction is proportional to the square of the velocity (friction type 1 in text Damping). The force should be then described as F = -k*x - k1*sign(v)*v2 Modify your model and test this hypothesis. x Form a hypothesis that this friction is proportional to the velocity (friction type 2) then the force should be described as F = -k*x - k1*v Modify your model and test this hypothesis. x Which hypothesis is correct? Why? Experiment The Result file Damped oscillations consists of data recorded during the following experiment. 62 The mass of 83 g was attached to a spring with 3.42 N/m. The mass of the spring was 15.0 g. (The mathematics shows that in the mass of the mass-spring system only 1/3 of the mass spring should be calculated). As a damper the thin cardboard was used (not shown on the picture). The damper was attached in such a way that the surface perpendicular to the direction of the motion was rather small. The measurement resolution in this experiment was 0.09 cm. Damping Damping is caused by friction. You can distinguish three types of friction: x Friction caused by collisions of a moving object against the air and liquid molecules. This one is proportional to the square of the velocity. Fr= -k1*sign(v)*v2 1) x Friction which is caused by dragging a layer of air or liquid around the moving object which is proportional to the velocity. 2) Fdrag= - k2*v x Friction caused by contact which can be described as Fc=-constant. To give the direction 3) Fc=-k3*sign(v) In all cases the direction of the friction force is opposite to the direction of motion (direction of the velocity). Different type of friction forces act in different situations. (Give some examples). 63 Activity: Cooling down a cup of coffee As soon as a hot cup of coffee is poured, it begins to cool. The rate of cooling is approximately proportional to the temperature difference between the object and its surroundings. This is known as Newton's law of cooling. In this activity you are going to investigate how fast does a cup of coffee cool down? Activity procedure 1. Open Coach 6 Activity Physics models > Cooling down a cup of coffee. 2. Analyze the model and perform assignments. Model Explanation In this model is assumed that there is a spontaneous heat transfer (through conduction, convection, radiation, and evaporation) from the coffee into surroundings, in a short time interval P=K*Tdifference. The K is the heat constant, where all four forms of heat transfer are included. The temperature difference between coffee and its surroundings is (Tc -T0). This difference determines the heat transfer. The heat transfer causes the decreasing of coffee temperature. The coffee temperature is calculated by Qc /(mc*c). Assignments 1. Execute the model. x On what factors depends the cooling process? x Determine from the graph how long it takes before the coffee is ready to drink (temperature 40°C). 2. Look at the model. On what factors depends the cooling process? x Is the cooling process linear? Explain. 3. Investigate, by using the Simulate option, what effect for the cooling process has a larger Kvalue. How can you increase K in practice? x Investigate by simulation how the cooling process depends on an amount of coffee. 4. Measure with a temperature sensor the cooling of the cup of coffee. Save the result of this experiment. x Load your measurement result as the background graph. x Try to adapt model parameters to fit the model into the experiment result. 5. Assume that instead of air the cup of coffee is placed in a water bath (0.5 l). The water bath will slowly warm up. x Adapt the model. Assume that all heat is used to warm up the water. x Investigate the rate of cooling in this situation. Does a cooling process goes faster or slower compare to the cooling in air? x The final bath temperature is higher then its surroundings temperature. This cooling process you can also add to your model. 64 Activity: A lamp connected to the capacitor Capacitors are used to store electrical energy. In this activity you are going to investigate how can you keep a lamp connected to a capacitor burning as long as possible? Activity procedure 1. Open Coach 6 Activity Physics models > A lamp connected to the capacitor. 2. Analyze the model and perform assignments. Model Explanation The model describes the discharging of a capacitor through the resistor R. The charge on the capacitor is changing in time with the rate of change dQ/dt which is equal to the generated current I. The initial charge value Q0 is calculated from of Q=C*U0 where U0 is initial potential of 5V. The actual potential difference across the capacitor is found from UC=Q/C. The current through the resistor is calculated from I=UC /R. The change of charge is found and the charge Q is then altered by this amount. Assignments 1. Execute the model. Read out when the potential difference reaches the 2.5V value. x What can you say about the rate of discharging of a capacitor? x Determine the RC-time constant from the graph. 2. How does the current change during the discharging of a capacitor? x Modify the diagram. Set the column C3 as the second vertical. x Determine the starting current value. x Draw a prediction of the current changing in the diagram. x Compare your prediction with the graph from the model 3. The process of discharging of a capacitor is determined by two circuit components: R and C. You are going to investigate how these values influence the discharging process. This can be done with the Simulate option. x Change only the R-value. Look how the discharging time changes. x Change then the C-value. Look how the discharging time changes. x Use the model to determine the resistance in a real circuit. Open the Result C=1000μF as a background graph. Use the Simulate option to determine what the R-value in this experiment was. x Use the Function Fit option to approximate (with mathematical functions) few of Vc(t) graphs (for different values of R and C). Select the most suitable function. Change the R and C values in the starting values of the model. - What can you say about the coefficient a? - Express the coefficient b in R*C. Change now also the UC value and do again the Function Fit. - Express the coefficient a in UC. . 65 Activity: The rocket Satellites are launched into a right earth orbit with the help of rockets. This is an expensive operation. It is good to use computer simulations to learn more about a behavior of a rocket in such situations. In this activity you will investigate how a satellite is launched into space orbit with a rocket? In a rocket, fuel is burned to make a hot gas. This hot gas is propelled out of the back of a rocket engine. The force of the gas in one direction (action) produces an opposing force (reaction) that propels that rocket forward. The 'Orbiting Solar Observatory' (OSO) was the first American space observatory (1962 - 1971) to study the sun activities. The OSO weighs 635 kg and had to revolve around Earth at a 563-km height. The OSO data are used as data for initial values used in the model. Activity procedure 1. Open Coach 6 Activity Physics models > The rocket. 2. Analyze the model and perform assignments. Model explanation The propulsion rocket model is already prepared in the Model window. This model describes the launching of the rocket with payload. The height of the rocket is changing in time. The rate of change of height is equal to the rocket velocity. The variable hk (=h/1000) is used to do height conversion from m to km. The velocity of the rocket is changing in time. The rate of change of velocity is equal to the rocket acceleration. The variable vk (=v/1000) is used to do velocity conversion from m/s to km/s. Acceleration is defined by the total force F divided by the total mass m. The total mass is = the satellite mass ms + the rocket mass mr+ + the fuel mass mf The fuel mass mf decreases in time with the rate of change dm/dt. When the fuel mass becomes smaller then zero that the fuel mass is set to zero. The fuel is burnt in a combustion chamber. Hot gas is propelled out of the back of a rocket engine. The force of the gas in one direction (action) produces an opposing force (reaction) that propels the rocket forward. 66 In the time interval dt an amount of fuel dm with speed c (relative to the rocket) is expelled from the rear of the rocket. The impulse exerted on this mass is: Fn*dt=-c*dm (direction upwards is positive). We assume that the speed of the gasses (relative to the rocket) remains constant. All forces occur in action - reaction pairs - 3rd Newton's law. In this case, the downward push on the exhaust gasses equals the upward push on the rocket. Fprop=-Fn Fprop=c*dm/dt. Two forces act on the rocket: propulsion force Fprop and the gravity force Fg. The total force F= the propulsion force Fprop - gravity force Fg In the model is assumed that all quantities are constant during the small time interval dt. For the first model: - This is one module rocket with liquid fuel. - The rocket is launched in vertical position. - The friction is neglected. - The gravity force is constant up to 563 km. Actually the assumption that the gravity force is constant is not correct. You can improve the model by changing the constant gravity Fg=m*g by the earth gravity Fg=G*me*m/(re+h)2, G the gravity constant (6.67e-11 Nm²/kg²); me the earth mass (5.976e24 kg); re the earth radius (6.38e6 m); h the height of the rocket path. Assignments 1. Study the model. Determine how long it takes before the fuel is used up. x Use the Sketch option to draw your prediction of using of the fuel mass mf in time. 2. The propulsion force acts on the rocket as long as there is a fuel. The rocket obtains its maximal speed of 3.24 m/s at time t = 200 s and reach the ground at time t = 911 s with end speed of -3.74 km/s. x Define forces that act on the rocket before the fuel is burnt up. x Use these data to make a prediction of the velocity - time graph starting at the moment 200 s. 3. Execute the model. x Read out the highest point, which is reached by the rocket and the time when it is happened. x Read out when the rocket reaches the ground. x What happens with the speed at t = 530 s? x Check with help of unit-analysis or the quantity [c*k] is a force. x Where can you find this quantity in the model? 4. At the moment the rocket reaches the correct height and velocity the control rockets turn it into the right orbit. This is not described in the model. x Calculate what velocity rocket has to get to reach orbit at 563-km height. 67 x Using the Simulate option determine the best value of the parameter c by which the rocket gets the desired speed and orbit height (at the same moment!) How much time it takes? TIP: near all values of v are used twice. 5. In the model is assumed that the gravity force is constant. Investigate where this assumption is valid. x Add to the model the earth gravity force so the model calculates values of x and v based on the earth gravity formulae. x Present both height-time graphs in one diagram. x Do the same for velocity-time graphs. x Execute the model and compare results of two forces. x Do you think that the assumption that the gravity force is constant is valid? 68

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