1 Astronauts on a distant planet toss a rock into the air. With the aid of a camera that takes pictures at a steady rate, they record the height of the rock as a function of time as given in the Table. (a) Make a position time graph. (b) Find the average velocity of the rock in the time interval between each measurement and the next. (c) Using these average velocities to approximate instantaneous velocities at the midpoints of the time intervals, make a graph of velocity as a function of time. (d) Does the rock move with constant acceleration? If so, plot a straight line of best fit on the graph and calculate its slope to find the acceleration 3 As two trains move along a track, their conductors suddenly notice that they are headed toward each other. The figure gives their velocities v as functions of time t as the conductors slow the trains. The slowing processes begin when the trains are 200 m apart. What is their separation when both trains have stopped? Answer 40 m 4 2 In one test to study neck injury in rear-end collisions, a volunteer was strapped to a seat that was then moved abruptly to simulate a collision by a rear car moving at 10.5 km/h. The figure gives the accelerations of the volunteer's torso and head during the collision, which began at time t = 0. The torso acceleration was delayed by 40 ms because during that time interval the seat back had to compress against the volunteer. The head acceleration was delayed by an additional 70 ms. What was the torso speed when the head began to accelerate? A particle moving along the positive x-axis has the following positions at various times: x(m) t(s) ∆x 0.089 0.0 +0.0 0.040 1.0 -0.049 0.010 2.0 -0.079 0.050 3.0 -0.039 0.080 4.0 -0.009 0.130 5.0 +0.041 0.200 6.0 +0.111 (a) Plot displacement (not position) versus time (b) find the average velocity of the particle in the intervals 0.0 to 1.0 s, 0.0 to 2.0 s, 0.0 to 3.0 s, 0.0 to 4.0 s (c) Find the slope of the curve drawn in part a at the points t = 0.0, 1.0, 2.0, 3.0, 4.0, and 5.0 s. (d) Plot the slope (units) versus time (e) From the curve of part (d), determine the acceleration of the particle at times t =2.0, 3.0, and 4.0 s. Answer (a) Graph (b) -.040 m/s -.035 m/s +.010 m/s +.0 m/s (c) Slope at t = 0 -.04 m/s Slope at t = 1 -.02 m/s Slope at t = 2 +.0 m/s Slope at t = 3 +.02 m/s Slope at t = 4 +04 m/s Slope at t = 5 +06 m/s (d) Graph (e) at t = 2 .02 m/s2 at t = 3 .02 m/s2 at t = 4 .02 m/s2 Answer 2 m/s 7.2 km/hr 5 Can an object have an eastward velocity while experiencing a westward acceleration? Answer Yes. Explain that such an object is slowing down. It will stop, then speed up toward the west. 6 The graph of x versus t in the figure is for a particle in straight line motion (a) State for each interval whether the velocity Vx is +, -, or 0, and whether the acceleration Ax is +, -, or 0. The intervals are OA, AB, BC, and CD. (b) From the curve, is there any interval over which the acceleration is obviously not constant? (Ignore the behavior at the end points of the intervals.) 8 A particle moves along the x-axis with a position versus time graph as shown in the figure. Sketch roughly curves of velocity versus time and acceleration versus time for this motion. Answer See graphs on answer page. Answer (a) OA AB BC CD Vx + + 0 - Ax 0 0 + 9 (b) no 7 The graph of x versus t in the figure is for a particle in straight line motion (a) State for each interval whether the velocity Vx is +, -, or 0, and whether the acceleration Ax is +, -, or 0. The intervals are OA, AB, BC, and CD. (b) From the curve, is there any interval over which the acceleration is obviously not constant? (ignore the behavior at the end points of the intervals.) For each of the following situations, sketch a velocity-tme graph that is a possible description of position as a function of time for a particle that moves along the x-axis. (a) At t = 1 s, the particle has zero velocity and positive acceleration; (b) At t = 1 s, the particle has zero velocity and negative acceleration; (c) At t = 1 s, the particle has negative velocity and positive acceleration; (d) At t = 1 s, the particle has negative velocity and negative acceleration (e) For which of these situations is the speed of the particle increasing at t = 1 s? Answer See answer page (e) For the speed to increase, the velocity and the acceleration must be in the same direction, which is the case for situations (a), (b) and (d) above. 10 Can the velocity of an object be zero at the same instant its acceleration is not zero? Give an example. Answer 11 Answer (a) Vx OA + AB 0 BC + CD + (b) no Ax 0 + 0 If one object has a greater speed than a second object, does the first necessarily have a greater acceleration? Explain, using examples. Answer 12 Can an object have a northward velocity and a southward acceleration? Explain. Answer 13 Can the velocity of an object be negative when its acceleration is positive? What about vice versa? 17 Answer 14 The figure shows the velocity of a train as a function of time. (a) At what time was its velocity greatest? (b) During what periods, if any, was the velocity constant? (c) During what periods, if any, was the acceleration constant? (d) When was the magnitude of the acceleration greatest? Give an example where both the velocity and acceleration are negative. Answer Answer 15 The position of a racing car, which starts from rest at t = 0 and moves in a straight line, has been measured as a function of time, as given in the table shown. (a) Draw a position time graph (b) Estimate its velocity as a function of time. (c) Estimate its acceleration as a function of time. 18 A high-performance automobile can accelerate approximately as shown in the velocity-time graph shown. (The jumps in the curve represent shifting of the gears.) (a) Estimate the average acceleration of the car when it is in first, (b) Estimate the average acceleration of the car when it is in third, (c) Estimate the average acceleration of the car when it is in fifth gear. (d) What is its average acceleration through the first four gears? Answer 16 Use the figure to answer the following questions: (a) During what time periods, if any, is the object's velocity constant? (b) At what time is its velocity the greatest? (c) At what time, if any, is the velocity zero? (d) Does the object run in one direction or in both along its tunnel during the time shown? Answer 19 Use the figure to answer the following questions: (a) Estimate the distance the object traveled during the first minute (b) Estimate the distance the object traveled during the second minute. Answer Answer 20 Construct the v vs. t graph for the object whose displacement as a function of time is given by the graph. 23 In a forward punch in karate, the fist begins at rest at the waist and is brought rapidly forward until the arm is fully extended. The speed v(t) of the fist is given in the figure for someone skilled in karate. (a) How far has the fist moved at time t = 50 ms? (b) How far has the fist moved when the speed of the fist is maximum? Answer Answer 21 The figure gives the velocity of a particle moving along an axis. Point 1 is at the highest point on the curve; point 4 is at the lowest point; and points 2 and 6 are at the same height. (a) What is the direction of travel at time t = 0? (b) What is the direction of travel at point 4? (c) At which of the six numbered points does the particle reverse its direction of travel? (d) Rank the six points according to the magnitude of the acceleration, greatest first. (a) 0.13 m; (b) 0.50 m 24 An iceboat has a constant velocity toward the east when a sudden gust of wind causes the iceboat to have a constant acceleration toward the east for a period of 3.0 s. A plot of x versus t is shown in the figure, where t = 0 is taken to be the instant the wind starts to blow and the positive x axis is toward the east. (a) What is the acceleration of the iceboat during the 3.0 s interval? (b) What is the velocity of the iceboat at the end of the 3.0 s interval? (c) If the acceleration remains constant for an additional 3.0 s, how far does the iceboat travel during this second 3.0 s interval? Answer (a) positive direction; (b) negative direction; (c) 3 and 5; (d) 2 and 6 tie, then 3 and 5 tie, then 1 and 4 tie (zero) 22 The figure depicts the motion of a particle moving along an x axis with a constant acceleration. (a) What is the magnitude of the particle's acceleration? (b) What is the direction of the particle's acceleration? Answer (a) 2.0 m/s2; (b) 12 m/s; (c) 45 m Answer (a) 4.0 m/s2; (b) +x 25 A graph of x versus t for a particle in straight-line motion is shown in the figure. (a) What is the average velocity of the particle between t = 0.50 s and t = 4.5 s? (b) What is the instantaneous velocity of the particle at t = 4.5 s? (c) What is the average acceleration of the particle between t = 0.50 s and t = 4.5 s? (d) What is the instantaneous acceleration of the particle at t = 4.5 s? 27 A particle starts from the origin at t = 0 and moves along the positive x axis. A graph of the velocity of the particle as a function of the time is shown in the figure. (a) What is the coordinate of the particle at t = 5.0 s? (b) What is the velocity of the particle at t - 5.0 s? (c) What is the acceleration of the particle at t = 5.0 s? (d) What is the average velocity of the particle between t = 1.0 s and t = 5.0 s? (e) What is the average acceleration of the particle between t = 1.0 s and t = 5.0 s? Answer (a) 15 m; (b) 2.0 m/s; (c) -2.0 m/s2; (d) 3.5 m/s; (e) 0 Answer (a) 2.5 m/s; (b) 8.0 m/s; (c) 1.0 m/s2; (d) 0 28 26 A graph of acceleration a versus t time for a particle as it moves along an x axis is shown in the figure. At t = 0 the coordinate of the particle is 4.0 m and the velocity v is 2.0 m/s. (a) What is the velocity of the particle at t = 2.0 s? (b) Write an expression for v(t) in meters per second that is valid for the interval 2.0 s < = t < = 4.0 s. The figure gives the acceleration a versus time t for a particle moving along an x axis. At t = -2.0 s, the particle's velocity is 7.0 m/s. What is its velocity at t = 6.0 s? Answer 39 m/s Answer (a) 5.0 m/s; (b) v = 3.0 m/s + (0.50 m/s3)t2 29 The graph of x versus t in the figure is for a particle in straight line motion Look at each interval. The intervals are OA, AB, BC, and CD. (a) State for each interval whether the velocity Vx is +, -, or 0 (b) State for each interval whether the acceleration Ax is +, -, or 0. (c) From the curve, is there any interval over which the acceleration is obviously not constant? (ignore the behavior at the end points of the intervals.) 31 A baseball pitcher throws a fast ball with a speed of 44 m/s. Estimate the average acceleration of the ball during the throwing motion. It is observed that in throwing the baseball, the pitcher accelerates the ball through a displacement of about 3.5 m from behind the body to the point where it is released (see the figure). Answer 276 m/s2 Answer 32 (a) Vx OA + AB 0 BC + CD + (b) Ax OA AB 0 BC + CD 0 (c) no 30 Of the following situations, which one is impossible? Explain your choice. (a) A body having velocity east and acceleration east (b) A body having velocity east and acceleration west. (c) A body having zero velocity but acceleration not zero. (d) A body having constant acceleration and variable velocity. (e) A body having constant velocity and variable acceleration. A car accelerates along a straight road from rest to 60 km/h in 5.0 s. (a) Draw a velocity-time graph of the motion (b) What is the magnitude of its average acceleration? Answer (a) graph (b) 3.33 m/s 33 Suppose a planner is designing an airport for small planes. One kind of airplane that might use this airfield must reach a speed before takeoff of 100 km/h (27.8 m/s) and can accelerate at 2.0 m/s2. If the runway is 150 m long, can this airplane reach the proper speed to take off? Answer 24.5 m/s Answer (e) is impossible since a non-zero acceleration means the velocity must be changing. Give examples of other situations. 34 How long does it take a car to travel 30.0 m if it accelerates from rest at a rate of 2.00 m/s2 Answer 5.48 s 35 A sprinter accelerates from rest to 10.0 m/s in 1.35 s. (a) Sketch a velocity-time graph. (b) What is her acceleration in m/s2? (c) What is her acceleration in km/h2? Answer (a) graph (b) 7.41 m/s2 (c) 9.6 x 104 km/hr 36 A sports car is advertised to be able to stop in a distance of 50 m from a speed of 90 km/h. (a) What is its acceleration in m/s2? (b) How many g's is this (g = 9.80 m/s2)? 42 A car traveling at 85 km/h strikes a tree; the front end of the car compresses and the driver comes to rest after traveling 0.80 m. What was the average deceleration of the driver during the collision? Express the answer in terms of "g's," where 1.00 g = 9.80 m/s2. Answer Answer (a) -3.5 x 102 m/s (b) -36 g’s 37 A car accelerates from 13 m/s to 25 m/s in 6.0 s. Assume constant acceleration. (a) What was its acceleration? (b) How far did it travel in this time? 43 Answer (a) 2.0 m/s2 (b) 114 m 38 Answer (a) 1.58 m/s2 (b) 3.52 s A light plane must reach a speed of 33 m/s for takeoff. How long a runway is needed if the (constant) acceleration is 3.0 m/s2? Answer 44 1.8 x 102 m 39 Spotting a police car, you brake a Porsche from a speed of 100 km/h to a speed of 80.0 km/h during a displacement of 88.0 m, at a constant acceleration. (a) What is that acceleration? (b) How much time is required for the given decrease in speed? A world-class sprinter can burst out of the blocks to essentially top speed (of about 11.5 m/s) in the first 15.0 m of the race. (a) What is the average acceleration of this sprinter (b) How long does it take her to reach that speed? At a certain time a particle had a speed of 18 m/s in the positive x direction, and 2.4 s later its speed was 30 m/s in the opposite direction. (a) Sketch a velocity-time graph (b) What is the average acceleration of the particle during this 2.4 s interval? Answer (a) graph (b) -20 m/s2 Answer (a) 4.41 m/s2 (b) 2.61 s 40 45 A car decelerates from a speed of 21.0 m/s to rest in 6.00 s. (a) Sketch a velocity-time graph (b) How far did it travel in that time? An electron has a constant acceleration of +3.2 m/s2. At a certain instant its velocity is +9.6 m/s. (a) What is its velocity 2.5 s earlier? (b) What is its velocity 2.5 s later? Answer (a) +1.6 m/s; (b) +18 m/s Answer (a) graph (b) 63 m 46 41 In coming to a stop, a car leaves skid marks on the highway 92 m long. Assuming a deceleration of 7.00 m/s2 (roughly the maximum for rubber tires on dry pavement), estimate the speed of the car just before braking. Answer 36 m/s An electric vehicle starts from rest and accelerates at a rate of 2.0 m/s2 in a straight line until it reaches a speed of 20 m/s. The vehicle then slows at a constant rate of 1.0 m/s2 until it stops. (a) How much time elapses from start to stop? (b) How far does the vehicle travel from start to stop? Answer (a) 30 s; (b) 300 m 47 A rocket-driven sled running on a straight, level track is used to investigate the physiological effects of large accelerations on humans. One such sled can attain a speed of 1600 km/h in 1.8 s, starting from rest. (a) Find the acceleration (assumed constant) in terms of g (b) Find the distance traveled. 52 A person driving her car at 45 km/h approaches an intersection just as the traffic light turns yellow. She knows that the yellow light lasts only 2.0 s before turning to red, and she is 30 m away from the near side of the intersection (see the figure). Should she try to stop, or should she make a run for it? The intersection is 12 m wide; her car's maximum deceleration is -5.8 m/s2, whereas it can accelerate from 45 km/h to 60 km/h in 6.0 s. Ignore the length of her car and her reaction time. Answer (a) 25 g; (b) 400 m 48 A certain sprinter has a top speed of 11.0 m/s. If the sprinter starts from rest and accelerates at a constant rate, he is able to reach his top speed in a distance of 12.0 m. He is then able to maintain this top speed for the remainder of a 100 m race. (a) What is his time for the 100 m race? (b) In order to improve his time, the sprinter tries to decrease the distance required for him to reach his top speed. What must this distance be if he is to achieve a time of 10.0 s for the race? Answer She should stop Answer (a) 10.2 s; (b) 10.0 m 49 53 A shuffleboard disk is accelerated at a constant rate from rest to a speed of 6.0 m/s over a 1.8 m distance by a player using a cue. At this point the disk loses contact with the cue and slows at a constant rate of 2.5 m/s2 until it stops. (a) How much time elapses from when the disk begins to accelerate until it stops? (b) What total distance does the disk travel? A car moving with constant acceleration covered the distance between two points 60.0 m apart in 6.00 s. Its speed as it passed the second point was 15.0 m/s. (a) What was the speed at the first point? (b) What was the magnitude of the acceleration? (c) At what prior distance from the first point was the car at rest? (d) Graph x versus t and v versus f for the car, from rest (t = 0). Answer (a) 5.00 m/s; (b) 1.67 m/s2; (c) 7.50 m Answer (a) 3.0 s; (b) 9.0 m 54 50 A jumbo jet must reach a speed of 360 km/h on the runway for takeoff. What is the lowest constant acceleration needed for takeoff from a 1.80 km runway To stop a car, first you require a certain reaction time to begin braking; then the car slows at a constant rate. Suppose that the total distance moved by your car during these two phases is 56.7 m when its initial speed is 80.5 km/h, and 24.4 m when its initial speed is 48.3 km/h. (a) What is your reaction time? (b) What is the magnitude of the acceleration? Answer Answer 2.78 m/s2 51 An automobile traveling 56 km/hr is 24.0 m from a barrier when the driver slams on the brakes. The car hits the barrier 2.0 seconds later .(give your answer in m/s2) (a) What was the automobile's deceleration before impact? (b) How fast was the car traveling at impact? (a) 0.74 s; (b) 6.2 m/s2 55 Make up a table of stopping distances for an automobile with an initial speed of 95 km/h and human reaction time of 1.0 s: (a) for an acceleration a = -4.0 m/s2; (b) for a = -8.0 m/s2. Answer (a) 3.56 m/s2 (b) 8.43 m/s Answer (a) 113 m (b) 70 56 A 90-m-long train begins accelerating uniformly from rest. The front of the train has a speed of 20 m/s when it passes a railway worker who is standing 180 m from where the front of the train started. What will be the speed of the last car as it passes the worker? (see the figure.) 60 The brakes on your car can slow you at a rate of 5.2 m/s2. (a) If you are going 137 km/h and suddenly see a state trooper, what is the minimum time in which you can get your car under the 90 km/h speed limit? (The answer reveals the futility of braking to keep your high speed from being detected with a radar or laser gun.) (b) Graph x versus t and v versus t for such a slowing. Answer (a) 2.5 s 61 Answer 31 m/s 57 A wombat moves along an x axis. (a) What is the sign of its acceleration if it is moving in the positive direction with increasing speed? (b) What is the sign of its acceleration if it is moving in the positive direction with decreasing speed? (c) What is the sign of its acceleration if it is moving in the negative direction with increasing speed? (d) What is the sign of its acceleration if it is moving in the negative direction with decreasing speed? Answer (a) 60.6 s; (b) 36.3 m/s 62 Answer (a) plus; (b) minus; (c) minus; (d) plus 58 Two subway stops are separated by 1100 m. A subway train accelerates at +1.2 m/s2 from rest through the first half of the distance and decelerates at -1.2 m/s2 through the second half. (a) What is its travel time? (b) What is its maximum speed? (c) Graph x, v, and a versus t for the trip. At t = 0, a particle moving along an x axis is at position x0 = -20 m. The signs of the particle's initial velocity v0 (at time t0 and constant acceleration a are, respectively, for four situations: (1) +, +; (2) +, -; (3) -, +; (4) -, -. (a) In which situations will the particle stop momentarily? (b) In which situations will the particle pass through the origin? (c) In which situations will the particle never pass through the origin? A car can be braked to a stop from the autobahn-like speed of 200 km/h in 170 m. (a) Assuming the acceleration is constant, find its magnitude in SI units. (b) Assuming the acceleration is constant, find its magnitude in in terms of g. (c) How much time T1, is required for the braking? (d) Your reaction time Tr is the time you require to perceive an emergency, move your foot to the brake, and begin the braking. If Tr = 400 ms, then what is T1, in terms of Tr ? (e) Is most of the full time required to stop spent in reacting or braking? (f) Dark sunglasses delay the visual signals sent from the eyes to the visual cortex in the brain, increasing Tr. In the extreme case in which Tr is increased by 100 ms, how much farther along the road does the car travel during your reaction time? Answer (a) 9.08 m/s2 (b) 0.926 g (c) 6.12 s (d) 15.34 to 1r (e) braking; (f) 5.56 m Answer (a) 2, 3; (b) 1, 3; (c) 4 63 59 On a dry road, a car with good tires maybe able to brake with a constant deceleration of 4.92 m/s2. (a) How long does such a car, initially traveling at 24.6 m/s, take to stop? (b) How far does it travel in this time? (c) Graph x versus t and v versus t for the deceleration. Most important in an investigation of an airplane crash by the U.S. National Transportation Safety Board is the data stored on the airplane's flight-data recorder, commonly called the "black box" in spite of its orange coloring and reflective tape. The recorder is engineered to withstand a crash with an average deceleration of magnitude 3400 g during a time interval of 6.50 ms. In such a crash, if the recorder and airplane have zero speed at the end of that time interval, what is their speed at the beginning of the interval? Answer Answer (a) 5.00 s; (b) 61.5 m 216 m/s 64 When a driver brings a car to a stop by braking as hard as possible, the stopping distance can be regarded as the sum of a "reaction distance," which is initial speed multiplied by the driver's reaction time, and a "braking distance,- which is the distance traveled during braking. The following table gives typical values. (a) What reaction time is the driver assumed to have? (b) What is the car's stopping distance if the initial speed is 25 m/s? 67 At the instant the traffic light turns green, an automobile starts with a constant acceleration a of 2.2 m/s2. At the same instant a truck, traveling with a constant speed of 9.5 m/s, overtakes and passes the automobile. (a) How far beyond the traffic signal will the automobile overtake the truck? (b) How fast will the automobile be traveling at that instant? Answer (a) 82 m; (b) 19 m/s 68 Answer (a) 0.75 s; (b) 50 m 65 Answer The maximum acceleration that is tolerable for a passenger in a subway train is 1.34 m/s2 and subway stations are located 806 meters apart. (a) What is the maximum speed a subway train could attain in the distance between stations? (b) What would be the time between stations? (c) If the subway train stops for a 20-s interval at each station, what is the maximum average speed of a subway train? Answer My Solution (a) 6 meters (b) 2 seconds (c) Particle A = 3 m/s Particle B = -4 m/s 69 (a) 32.9 m/s (b) 49.1 seconds (c) 11.7 m/s 66 A car moving with constant acceleration covers the distance between two points 60 m apart in 6.00 s. Its speed as it passes the second point is 15 m/s: (a) What is its speed at the first point? (b) What is its acceleration? (c) At what prior distance from the first point was the car at rest? (d) Graph x versus t for the trip (e) Graph a v versus t for the trip. (f) Graph an a versus t for the trip. A particle is moving in the direction of the positive x-axis with a constant speed of 3.0 m/s. As it passes the origin, a second particle initially at rest at x = 10 m begins accelerating toward the origin with a constant acceleration whose magnitude is 2.0 m/s2 (a) Where do the two particles meet? (b) When do the two particles meet? (c) What are the velocities of the two particles just before they collide? Two trains, one traveling at 72 km/h and the other at 144 km/h are headed toward one another along a straight level track. When they are 950 meters apart, both engineers simultaneously see the other's train and apply their breaks. If the brakes decelerate each train at the rate of 1.0 m/s2 determine if there is a collision. Answer Collision 70 The driver of a blue car (80 km/h) suddenly realizes that she is about to rear-end a red car (60 km/h). To avoid a collision, what is the maximum speed the blue car can have just as it reaches the red car? Answer Answer My Solution (a) vi = 5 m/s (b) a = 1.67 m/s2 (c) d = 7.48 m (d) (e) (f) 60 km/h, not 0 71 At t = 0 and x = 0, an initially stationary blue car begins to accelerate at the constant rate of 2.0 m/s2 in the positive direction of the x axis. At t = 2 s, a red car traveling in an adjacent lane and in the same direction passes x = 0 with a speed of 8.0 m/s and a constant acceleration of 3.0 m/s2. What pair of simultaneous equations should be solved to find when the red car passes the blue car? Answer x = t2 and x = 8(t - 2) + 1.5(t - 2)2, with x in meters and t in seconds 72 Two trains, each having a speed of 30 km/h, are headed at each other on the same straight track. A bird that can fly 60 km/h flies off the front of one train when they are 60 km apart and heads directly for the other train. On reaching the other train, the bird flies directly back to the first train, and so forth. (We have no idea why a bird would behave in this way.) What is the total distance the bird travels before the trains collide? 74 Answer 60 km 73 When a high-speed passenger train traveling at 161 km/h rounds a bend, the engineer is shocked to see that a locomotive has improperly entered onto the track from a siding and is a distance D = 676 m ahead (see the figure). The locomotive is moving at 29.0 km/h. The engineer of the high-speed train immediately applies the brakes. (a) What is the time to slow down? (b) What must be the magnitude of the resulting constant deceleration if a collision is to be just avoided? (c) Assume that the engineer is at x = 0 when, at t = 0, he first spots the locomotive. Sketch x(t) curves for the locomotive and high-speed train for the cases in which a collision is just avoided and is not quite avoided. In one test to study neck injury in rear-end collisions, a volunteer was strapped to a seat that was then moved abruptly to simulate a collision by a rear car moving at 10.5 km/h. The figure gives the accelerations of the volunteer's torso and head during the collision, which began at time t = 0. The torso acceleration was delayed by 40 ms because during that time interval the seat back had to compress against the volunteer. The head acceleration was delayed by an additional 70 ms. (a) At maximum head acceleration, what is the speed of the head and (b) At maximum head acceleration, what is the speed of the torso? Answer (a) 2.25 m/s; (b) 3.90 m/s 75 When a soccer ball is kicked toward a player and the player deflects the ball by "heading" it, the acceleration of the head during the collision can be significant. The figure gives the measured acceleration a(t) of a soccer player's head for a bare head and a helmeted head, starting from rest. At time t = 7.0 ms, what is the difference in the speed acquired by the bare head and the speed acquired by the helmeted head? Answer (a) 36.8 s (b) 0.994 m/s2 (c) see Graph Answer 0.56 m/s 76 A red train traveling at 72 km/h and a green train traveling at 144 km/h are headed toward each other along a straight, level track. When they are 950 m apart, each engineer sees the other's train and applies the brakes. The brakes slow each train at the rate of 1.0 m/s2. Is there a collision? If so, answer yes and give the speed of the red train and the speed of the green train at impact, respectively. If not, answer no and give the separation between the trains when they stop. Answer yes, 0, 10 m/s 77 The following equations give the position x(t) of a particle in four situations (in each equation, x is in meters, t is in seconds, and t > 0): (1) x = 3t2; (2) x = -4t2 - 2; (3) x = 2/t2; and (4) x = -2. (a) In which situation is the velocity v of the particle constant? (b) In which is v in the negative x direction? 80 Consider the two quantities (dx/dt)2 and d2x/dt2 (a) Are these merely two equivalent expressions for the same thing? (b) What are the SI units of these two quantities? Answer (a) No (b) m2/s2 m/s2 Answer (check the derivative dx/dt) (a) 4; (b) 2 and 3 81 78 3 The position of an object is given by x = 2t , where x is measured in meters and t in seconds. (a) Find the instantaneous velocities and the instantaneous accelerations at t = 1 s and t = 2 s (b) Find the average velocity and the average acceleration between t = 1 s and t = 2 s Answer (a) v(2) = 24 m/s v(1) = 6 m/s a(2) = 24 m/s2 a(1) = 12 m/s2 (b) 14 m/s 18 m/s2 79 The position of a particle moving along the x-axis depends on the time according to the equation: x = at2 - bt3 where x is in meters and t in seconds. (a) What dimensions and units must a and b have? For the following parts let their numerical values be 3 and 1 respectively. (b) At what time does the particle reach its maximum positive x-position? (c) What total length of path does the particle cover in the first 4 s? (d) What is its displacement during the first 4 s? (e) What is the particle's velocity at the end of each of the first four seconds? (f) What is the particle's acceleration at the end of each of the first four seconds? Answer (a) L/T2 m/s2 m/s3 (b) 2s (c) 24 m (d) -16 m (e) +3 m/s +0 m/s -9 m/s -24 m/s (f) +0 m/s2. -6 m/s2. -12 m/s2. -18 m/s2. The position of a particle as it moves along the x axis is given by x = 15e-t m, where t is in seconds (a) What is the position of the particle at t = 0, 0.50, and 1.0 s? (b) What is the average velocity of the particle between t = 0 and t = 1.0 s? (c) What is the instantaneous velocity of the particle at t = 0? (d) What is the instantaneous velocity of the particle at t = 0.50? (e) What is the instantaneous velocity of the particle at t = 1.0 s? (f) Plot x versus t for the particle for 0< = t < = 1.0 s and estimate the instantaneous velocity at t = 0.50 s from the graph. Answer (a) 15 m 9.1 m 5.5 m (b) -9.5 m/s (c) -15 m/s (d) -9.1 m/s (e) -5.5 m/s (f) -11 m/s 82 The position of a particle as it moves along the y axis is given by y = 2.0 sin ((π/4)t) where t is in seconds, y is in centimeters and the angle is in radians. (a) What is the average velocity of the particle between = 0 and t = 2.0 s? (b) What is the instantaneous velocity of the particle at t = 0, 1.0 and 2.0 s? (c) What is the average acceleration of the particle between t = 0 and t = 2.0 s? (d) What is the instantaneous acceleration of the particle at t = 0, 1.0 and 2.0 s? (e) Plot v versus t for 0< = t < = 2.0 s and estimate the instantaneous acceleration at t = 1.0 s from the graph. Answer (a) 1.0 cm/s (b) 1.6 cm/s 1.1 cm/ 0 (c) -0.79 cm/s2 (d) 0 -0.84 cm/s2 83 The position of a particle moving on an x axis is given by the function: x = 7.8 + 9.2t - 2.1t3, with x in meters and t in seconds. (a) What is its velocity at t = 3.5 s? (b) Is the velocity constant, or is it continuously changing? 86 Answer (a) -68 m/s (b) coninuously changing 84 A particle's position is given by x = 4 - 12t + 3t2 (where t is in seconds and x is in meters), (a) What is its velocity at t = 1 s? (b) Is it moving in the positive or negative direction of x just then? (c) What is its speed just then? (d) Is the speed increasing or decreasing just then? (e) Is there ever an instant when the velocity is zero? If so, give the time t; if not, answer no. (f) Is there a time after t = 3 s when the particle is moving in the negative direction of x? If so, give the time t; if not, answer no. Answer The following equations give the position x(t) of a particle in four situations: (1) x = 3t - 4; (2) x = -5t3 + 4t2 + 6; (3) x = 2/t2 4/t; (4) x = 5t2 - 3. To which of these situations do the equations of the table apply? (a) -6 m/s; (b) -x direction; (c) 6 m/s; (d) decreasing; (e) 2 s; (f) no 87 The position of a particle moving along the x axis is given in centimeters by x = 9.75 + 1.50t3, where t is in seconds. (a) Calculate the average velocity during the time interval t = 2.00 s to t = 3.00 s; (b) Calculate the instantaneous velocity at t = 2.00 s. (c) Calculate the instantaneous velocity at t = 3.00 s. (d) Calculate the instantaneous velocity at t = 2.50 s. (e) Calculate the instantaneous velocity when the particle is midway between its positions at t = 2.00 s and t = 3.00 s. (f) Graph x versus t and indicate your answers graphically. Answer Answer 1 and 4 a = d2x/dt2 must be a constant 85 The position function x(t) of a particle moving along an x axis is x = 4.0 - 6.0t2, with x in meters and t in seconds. (a) At what time oes the particle (momentarily) stop? (b) Where does the particle (momentarily) stop? (c) At what negative time does the particle pass through the origin? (d) At what positive time does the particle pass through the origin? (e) Graph x versus t for the range -5 s to +5 s. (f) To shift the curve rightward on the graph, should we include the term +20t or the term -20t in x(t)? (g) Does that inclusion increase or decrease the value of x at which the particle momentarily stops? (a) 28.5 cm/s; (b) 18.0 cm/s; (c) 40.5 cm/s; (d) 28.1 cm/s; (e) 30.3 cm/s 88 The position of a particle is given by x = 20t - 5t3, where x is in meters and t is in seconds. (a) When, if ever, is the particle's velocity zero? (b) When is its acceleration a zero? (c) For what time range (positive or negative) is a negative? (d) For what time range (positive or negative) is a positive? (e) Graph x(t), v(t), and a(t). Answer Answer (a) 0; (b) 4.0m; (c) -0.82 s; (d) 0.82 s; (f) +20t; (g) increase (a) 1.2 s; (b) 0; (c) positive; (d) negative 89 The position of a particle moving along an x axis is given by x = 12t2 - 2t3, where x is in meters and t is in seconds. (a) Determine the position of the particle at t = 3.0 s. (b) Determine the velocity of the particle at t = 3.0 s. (c) Determine the acceleration of the particle at t = 3.0 s. (d) What is the maximum positive coordinate reached by the particle? (e) At what time is the maximum positive coordinate reached by the particle? (f) What is the maximum positive velocity reached by the particle? (g) at what time is the maximum positive velocity reached by the particle? (h) What is the acceleration of the particle at the instant the particle is not moving (other than at t = 0)? (i) Determine the average velocity of the particle between t = 0 and t = 3 s. 92 Answer (a) 80 m/s; (b) 110 m/s; (c) 20 m/s2 Answer (a) 54 m; (b) 18 m/s; (c) -12 m/s2; (d) 64 m; (e) 4.0 s; (f) 24 m/s; (g) 2.0 s; (h) -24 m/s2; (i) 18 m/s 90 In an arcade video game, a spot is programmed to move across the screen according to x = 9.00t - 0.750t3, where x is distance in centimeters measured from the left edge of the screen and t is time in seconds. When the spot reaches a screen edge, at either x = 0 or x = 15.0 cm, t is reset to 0 and the spot starts moving again according to x(t). (a) At what time aftcr starting is the Spot instantaneously at rest? (b) At what value ofx does this occur? (c) What is the spot's acceleration (including sign) when this occurs'? (d) Is it moving right or left just prior to coming to rest? (e) Is it moving right or left coming to rest? (f) (optional) At what time t > 0 does it first reach an edge of the screen? 93 91 The acceleration of a particle along an x axis is a = 5.0t, with t in seconds and a in meters per second squared. At t = 2.0 s, its velocity is +17 m/s. What is its velocity at t = 4.0 s? Answer A motorcyclist who is moving along an x axis directed toward the east has an acceleration given by a = (6.1 - 1.2t) m/s2 for 0 < = t < = 6.0 s. At t = 0, the velocity and position of the cyclist are 2.7 m/s and 7.3 m. (a) What is the maximum speed achieved by the cyclist? (b) What total distance does the cyclist travel between t = 0 and 6.0 s? Answer (a) 18 m/s; (b) 83 m 94 A Particle moves along the x-axis according to the equation x = 50t + 10t3 where x is in meters and t is in seconds. (a) Calculate the instantaneous velocity of the particle at t = 3.0 s, and (b) Calculate the instantaneous acceleration of the particle at t = 3.0 s. (c) Calculate the average velocity of the particle during the first 3 s of its motion Answer Answer (a) 2.00 s; (b) 12 cm; (c) -9.00 cm/s2; (d) right; (e) left; (f) 3.46 s A proton moves along the x axis according to the equation x = 50t + 10t2, where x is in meters and t is in seconds. (a) Calculate the average velocity of the proton during the first 3.0 s of its motion. (b) Calculate the instantaneous velocity of the proton at t = 3.0 s. (c) Calculate the instantaneous acceleration of the proton at t = 3.0 s. (d) Graph x versus t and indicate how the answer to (a) can be obtained from the plot. (e) Indicate the answer to (b) on the graph. (f) Plot v versus t and indicate on it the answer to (c). (a) 320 m/s (b) 180 m/s2 (c) 140 m/s 95 Answer the following questions (a) If a particle's position is given by x = 4 + 12t + 3t2 where t is in seconds and x s in meters. (a) what is its velocity at t = 1 s? (b) Is it moving toward increasing or decreasing x at t = 1 s? (c) What is its speed at t = 1 s? (d) Is the speed larger or smaller at later times? (Try answering the next two questions without further calculations. (e) Is there ever an instant when the velocity is zero? (f) Is there a time after t = 3 s when the particle is moving leftward on the x axis? +47 m/s Answer (a) (b) (d) (e) (f) = 6 m/s negative x direction. 6 m/s first smaller then zero and then larger yes (t = 2 s) no 96 Answer the following questions (a) If the position of a particle is given by the equation x = 20 t - 5t3, where x is in meters and t is in seconds, when if ever, is the particle's velocity zero? (b) When is its acceleration zero? (c) When is the acceleration negative (d) When is the acceleration Positive? (e) Graph x(t), (f) Graph v(t) (g) Graph a(t) 100 Describe in words the motion of the object graphed in the figure Answer Answers (a) 5= 1,2 s (b) t = 0 (c) t > 0 t<0 97 A bus accelerates at 1.5 m/s2 from rest for 12 s. It then travels at a constant velocity for 25 s, after which it slows to a stop with an acceleration of -1.5 m/s2 (a) how far does the bus travel? (b) what is its average velocity? 98 A car manufacturer tests its cars for front-end collisions by hauling them up on a crane and dropping them from a certain height. (a) Derive the quadratic bypass from the slope formula and the displacement formula. (b) What height corresponds to a collision at 50 km/h? (c) What height corresponds to a collision at 100 km/h? Initially, the object moves in the positive direction with a constant acceleration, until about t = 45 s, when it has a velocity of (you need to finish the analysis) 101 The position of a rabbit along a straight tunnel as a function of time is plotted in the figure. What is its instantaneous velocity (a) What is its instantaneous velocity at t = 10.0 s? (b) What is its instantaneous velocity at t = 30.0 s? (c) What is its average velocity between t = 0 and t = 5.0 s? (d) What is its average velocity between t = 25.0 s and t = 30.0 s? (e) What is its average velocity between t = 40.0 s and t = 50.0 s? Answer (a) (b) 9.84 m (c) 39 m 99 Answers (a) .284 m/s (b) 1 m/s +/- .5 m/s (c).294 m/s (d) 1.4 m/s (e) -.8 m/s Describe in words the motion plotted in the figure. 102 Construct the x vs. t graph for the object whose velocity as a function of time is given by the graph shown. Answers The object starts with a constant velocity in the positive direction. at about t =17 s, when the object is at 5h3 5 meter position, it begins to gain speed--it has a positive acceleration. at about 5 = 27 s, when the object is about the 12 m position, it begins to slow down--it has a negative acceleration. The object instantaneously stops at about t = 37 s, reaching its maximum distance from the origin of 20 m. The object then reverses direction, gaining speed while moving backwards. At about t= 47 s, when the object is again at about the 12 m position, the object starts to slow down, and appears to stop at t=50 s, 10 m from the starting point. Answers 103 A graph of acceleration a versus t time for a particle as it moves along an x axis is shown in the figure. At t = 0 the coordinate of the particle is 4.0 m and the velocity v is 2.0 m/s. (a) What is the change in velocity of the particle at t = 1.0 s? (b) What is the change in velocity of the particle at t = 2.0 s? (c) What is the change in velocity velocity of the particle at t = 4.0 s? 107 Sketch a velocity-time graph for a shuttle train which runs between cities A and C with an intermediate stop at city B. All cities are on the same line. Answers 108 The graph shows the position of a radio controlled plane. (a) Construct a table showing the average velocity of the plane during each 10 second interval over the entire 100 seconds. (b) Plot a velocity-time graph using the table you constructed in part a. Answers (a) 1 m/s (b) 3.0 m/s (c) 9.0 m/s 104 105 A bowling ball with a negative initial velocity slows down as it rolls down the lane toward the pins. Is the bowling ball's acceleration positive or negative as it rolls toward the pins? Answers Answers positive (a) (b) (c) (d) (e) Two cars are moving in the same direction in parallel lanes along a highway. At some instant, the instantaneous velocity of car A exceeds the instantaneous velocity of car (B) Does this mean that car A's acceleration is greater than car B's? Explain, and use examples. 109 10 m/s 0 m/s -20 m/s -10 m/s -20 m/s Sketch a velocity-time graph starting at x = 0 and t = 0 having the velocities 10 m/s during the first time interval t = 0 to t = 1 s and 5 m/s during a second time interval of t = 1 to t = 2 s. Answers No, car A’s acceleration is not necessarily greater than car B’s acceleration. If the two cars are moving in the positive direction, car a could be slowing down (negative acceleration) while car b is speed up (positive acceleration), even though car A’s velocity is greater than car B’s velocity. 106 Answers 110 The figure shows a velocity-time graph for an automobile on a test track. Describe the changes in velocity with time. Sketch the velocity-time graphs for the following motions. (a) a city bus moving with a constant velocity (b) a wheelbarrow speeding up at a uniform rate moving in the positive direction (c) a tiger speeding up at a uniform rate moving in the negative direction (d) an iguana slowing down at a uniform rate moving in the positive direction (e) a camel slowing down at a uniform rate moving in the negative direction Answers (a) (b) (c) (d) (e) slope is zero slope is positive slope is negative slope is negative slope is positive Answers The car starts from rest and increases its speed. As the car's speed increases the driver shifts gears 111 Plot a velocity-time graph using the information in the table. 113 The strobe photographs in the figure show a disk moving from left to right under different conditions. The time interval between images is constant. Assuming that the direction to the right is positive, identify the following types of motion in each photograph (a) the acceleration is positive (b) the acceleration is negative (c) The velocity is constant Answers 112 The velocity-versus-time graph for a shuttle bus moving along a straight path is shown in the figure below. (a) Identify the time intervals during which the velocity of the shuttle bus is constant. (b) Identify the time intervals during which the acceleration of the shuttle bus is constant. (c) Find the value for the average velocity of the shuttle bus during each time interval identified in (d) Find the acceleration of the shuttle bus during each time interval identified in (e) Identify the times at which the velocity of the shuttle bus is zero. (f) Identify the times at which the acceleration of the shuttle bus is zero. (g) Explain what the shape of the graph reveals about the acceleration in each time interval Answers (a) a (left half) c (b) a (right half) (c) b 114 You drive on Interstate 10 from San Antonio to Houston, half the time at 55 km/h and the other half at 90 km/h. On the way back you travel half the distance at 55 km/h and the other half at 90 km/h. (a) What is your average speed from San Antonio to Houston? (b) What is your average speed from Houston back to San Antonio? (c) What is your average speed for the entire trip? (d) What is your average velocity for the entire trip? (e) Sketch x versus t for (a), assuming the motion is all in the positive x direction. Indicate how the average velocity can be found on the sketch. Answer (a) (b) (c) (d) 0 s to 30 s 60 s to 125 s 210 s to 275 s 0 s to 30 s 30 s to 60 s 60 s to 125 s 125 to 210 s 210 s to 275 s 275 s to 300 s 300 s to 520 s 520 s to 580 s 0 m/s 1.5 m/s 0 m/s -0.75 m/s -3.25 m/s -4.5 m/s 0 m/s2 0.1 m/s2 0 m/s2 -0.04 m/s2 0 m/s2 -0.06 m/s2 (a) 73 km/h; (b) 68 km/h; (c) 70 km/h; (d) 0 115 You are to drive to an interview in another town, at a distance of 300 km on an expressway. The interview is at 11:15 A.M. You plan to drive at 100 km/h, so you leave at 8:00 A.M. to allow some extra time. You drive at that speed for the first 100 km, but then construction work forces you to slow to 40 km/h for 40 km. What would be the least speed needed for the rest of the trip to arrive in time for the interview? Answer 128 km/h 116 When a soccer ball is kicked toward a player and the player deflects the ball by "heading" it, the acceleration of the head during the collision can be significant. The figure gives the measured acceleration a(t) of a soccer player's head for a bare head and a helmeted head, starting from rest. (a) What is the velocity of the bare head at t = 7.0 ms (b) What is the velocity of the helmeted head at t = 7.0 m/s (c) What is the difference in the speed acquired by the bare head and the speed acquired by the helmeted head at t = 7.0 ms? 118 The figure shows the velocity of an object is plotted against time. (a) What is the acceleration at t = 1.4 seconds? (b) What is the acceleration at t = 2.4 seconds? (c) What is the acceleration at t = 3.4 seconds? Answers (a) -5.71 m/s2 to -4 m/s2 (b) -1.82 m/s2 (c) 0 m/s2 to .5 m/s2 Answers (a) .82 m/s (b) .26 m/s (c) 0.56 m/s 117 119 An object moves at the speed shown in the graph found in the figure. Calculate the object's acceleration at t = 4 seconds. The graph shows the motion of an object over a given time period. (a) Graphically determine the slope of the curve at 2 second. (b) Graphically determine the slope of the curve at 4 seconds. Answers -10 m/s2 Answers (a) 5 m/s2 (b) 2 m/s2 120 An object moves at the speed shown in the graph found in the figure. (a) Use the graph to determine the instantaneous acceleration at 2 second. (b) Use the graph to determine the instantaneous acceleration at 4 seconds. (c) Use the graph to determine the instantaneous acceleration at 7 seconds. (d) Use the graph to determine the instantaneous acceleration at 9 seconds. 122 The figure shows the velocity of an object is plotted against time. What is the average acceleration of the object during the interval from 2 seconds to 6 seconds. Answers 4.5 m/s2 123 Answers For the graph found in the figure, determine the acceleration between 2 and 5 seconds (a) +6 m/s2 (b) +1.5 m/s2 (c) -2.75 m/s2 (d) -8.33 m/s2 121 An object moves at the speed shown in the graph found in the figure. (a) Use the graph to determine the instantaneous acceleration at 2 second. (b) Use the graph to determine the instantaneous acceleration at 4 seconds. (c) Use the graph to determine the instantaneous acceleration at 18 seconds. (d) Use the graph to determine the instantaneous acceleration at 10 seconds. Answers 3.67 m/s2 124 As a shuttle bus comes to a normal stop, it slows from 9.00 m/s to 0.00 m/s in 5.00 s. (a) Draw a velocity-time graph of the motion (b) Find the average acceleration of the bus. Answers (a) graph 9b) -1.80 m/s2 Answers (a) 3.75 m/s2 (b) 5 m/s2 (c) 4.27 m/s2 (d) 2.67 m/s2 125 When the shuttle bus comes to a sudden stop to avoid hitting a dog, it slows from 9.00 m/s to 0.00 m/s in 1.50 s. (a) Sketch a velocity time graph of the motion of the shuttle bus. (b) Use the graph to find the average acceleration of the bus. Answers (a) (b) -6.00 m/s2 126 A car traveling initially at 7.0 m/s accelerates to a velocity of 12.0 m/s in 2.0 s. What is the average acceleration of the car? (a) Sketch a velocity time graph of the motion of the car. (b) Use the graph to find the average acceleration of the car. 130 Answers (a) (b) +2.5 m/s2 127 Turner's treadmill starts with a velocity of -1.2 m/s and speeds up at regular intervals during a half-hour workout. After 25 min, the treadmill has a velocity of -6.5 m/s. (a) Sketch a graph of the situation described above. (b) What is the average acceleration of the treadmill during this period? The velocity-versus-time graph for a shuttle bus moving along a straight path is shown in the figure. (a) Identify the time intervals during which the velocity of the shuttle bus is constant. (b) Identify the times during which the acceleration of the shuttle bus is constant. (c) find the value for the average velocity of the shuttle bus during each time interval identified in b. (d) Find the acceleration of the shuttle bus during each time interval identified in b. (e) Identify the times at which the velocity of the shuttle bus is zero. (f) identify the times at which the acceleration of the shuttle bus is zero. (g) explain what the slope of the graph reveals about the acceleration in each time interval. Answers (a) graph (b) -3.53 x 10-3 m/s2 128 A treadmill starts at a velocity of -2.7 m/s and has a velocity of -1.3 m/s after 5.0 min. (a) Draw a velocit-time graph of the motion of the person on the hreadmill. (b) What is the average acceleration of the treadmill? (a) 0 s to 30 s 60 s to 125 s 210 s to 275 s Answers (a) graph (b) +4.7 x 10-3 m/s2 129 (b) 0 s to 30 s 30 s to 60 s 60 s to 125 s 125 to 210 s 210 s to 275 s 275 s to 300 s 300 s to 520 s 520 s to 580 s An object moves at the speed shown in the graph found in the figure. Find the average acceleration between 4 and 10 seconds. (c) 0 m/s 1.5 m/s 0 m/s -0.75 m/s -3.25 m/s -4.5 m/s (d) 0 m/s2 0.1 m/s2 0 m/s2 131 Answers 4.5 m/s2 The velocity-time graph for an object moving along a stright path is shown in the figure. Find the average accelerations during the time intervals (a) 0.0 to 5.0 s (b) 5.0 to 15.0 s (c) 0.0 to 15.0 s Answers (a) 0.0 m/s2 (b) +1.36 m/s2 (c) +0.68 m/s2 132 Velocity can be either positive or negative, depending on the direction of the displacement. The time interval, ∆t, is always positive. Why? 135 Answers the time interval is always positive because time can only move in one direction (forward) 133 Hanging over the railing of a bridge, you drop an egg (no initial velocity) as you throw a second egg downward. (a) Which curves in the figure give the velocity v(t) for the dropped egg? (b) Which curves in the figure give the velocity v(t) for the thrown egg? (Curves A and B are parallel; so are C, D, and E; so are F and G.) The figure shows the motion of a moving object. (a) Find the acceleration of the object during the first 5 seconds of travel. (b) Find the acceleration of the object during the second 5 seconds of travel. (c) Find the acceleration of the object between the tenth and the fifteenth second of travel. (d) Find the acceleration of the object between the twentieth and twenty-fifth second of travel. Answers (a) D; (b) E 136 With an average acceleration of -0.50 m/s2, how long will it take a cyclist to bring a bicycle with an initial velocity of +13.5 m/s to a complete stop? (Sketch a velocity time graph to help you calculate your answer Answers (a) (b) (c) (d) Answers 6.0 m/s2 0 m/s2 -2 m/s2 -4 m/s2 27 s 137 134 The figure shows a velocity-time graph of a toy train. (a) During which time interval or intervals is the train’s speed constant? (b) During which interval or intervals is the train's acceleration positive? (c) During which interval or intervals is the train’s acceleration less than zero? (d) During which time interval is the train’s acceleration most negative? A golf ball rolls up a hill toward a Putt-Putt hole. (a) If it starts with a velocity of + 2.0 m/s and accelerates at a constant rate of -0.50 m/s2, what is its velocity after 2.0 s? (b) If the acceleration occurs for 6.0 s, what is its final velocity? (c) Describe, in words, the motion of the golf ball. Answers (a) 1.0 m/s (b) -1.0 m/s (c) Velocity decreases the ball rolls back downhill. The ball is hit up hill and then rolls back down the hill. 138 To qualify for the finals in a racing event, a race car must achieve an average speed of 250 km/h on a track with a total length of 1600 m. (a) If a particular car covers the first half of the track at an average speed of 230 km/h, what minimum average speed must it have in the second half of the event to qualify? (b) Show how you would solve this problem using a graph. Answer Answers (a) 5 to 15 20 to 25 (b) 0 to 5 s (c) 15 to 20 s 25 to 40 s (d) 15 to 20 s 280 km/r 139 A tortoise can run with a speed of 10.0 cm/s, and a hare can run exactly 20 times as fast. In a race, they both start at the same time, but the hare stops to rest for 2.00 min. The tortoise wins by 20.0 cm. (a) Draw a position time graph for both the tortoise and the hair. Put both critters on the same axis. (b) How long does the race take? (c) What is the length of the race? (d) Solve the problem using algebra. 142 In the figure shown the velocity of an object is plotted against time. What is the average acceleration between 1 and 3 seconds? Answer (a) 126 s (b) 1260 cm 140 Runner A is initially 6.0 km west of a flagpole and is running with a constant velocity of 9.0 km/h due east. Runner B is initially 5.0 km east of the flagpole and is running with a constant velocity of 8.0 km/h due west. (a) Draw a position time graph for the runners (b) How far are the runners from the flagpole when their paths cross? (c) Solve the problem by algebra. Answers -3.75 m/s2 143 For the graph found in the figure, determine the average acceleration between 1.5 and 4.5 seconds. Answer 0.2 km west of the flagpole 141 Given the Following Table: (a) Plot a velocity-time graph for this motion. (b) Is this motion constant velocity? (c) Does the graph show uniform acceleration? (d) Calculate the instantaneous acceleration at 3 seconds. TIME (S) VELOCITY (M/S) 0.0 1.0 2.0 3.0 4.0 0.0 5.0 20.0 45.0 80.0 Answers 144 An object moves at the speed shown in the graph found in the figure. Find the average acceleration between 1 and 9 seconds Answers (a) (b) (c) (d) Parabolic no no Acceptable range 25 to 30 m/s2 Answers 145 In 1994, a human-powered submarine was designed in Boca Raton, Florida. It achieved a maximum speed of 3.06 m/s Suppose this submarine starts from rest and accelerates at 0.800 m/s2 until it reaches maximum speed. The submarine then travels at constant speed for another 5.00 s. (a) Draw a graph of the motion of the submarine (b) Calculate the total distance traveled by the submarine. 148 Answers (a) graph (b) 21.2 m 146 TIME (SEC) O 1 2 3 4 5 6 7 8 VELOCITY 1 4 8 12 16 20 20 20 20 Use graphing to solve the following problem. At the moment car A is starting from rest and accelerating at 4.0 m/s2, car B passes it moving at a constant speed of 28 m/s. How long will it take car A to catch car B? Answers (a) (b) (c) (d) (e) (f) Answers 14 seconds 147 The graph show the motion of an object during a 25 second time interval. (a) What displacement did the object make between t = 0 and t = 5 seconds. (b) What displacement did the object make between t = 5 seconds and t = 10 seconds. (c) What displacement did the object make between t = 10 and t = 15 seconds. (d) What displacement did the object make between t = 0 and t = 25 seconds. (e) Sketch a position time graph of the object’s motion (f) Sketch an acceleration time graph of the object’s motion. Make sure the time’s of all three of your graphs line up. The velocity of an automobile changes over an 8 second time period as shown in the table. (a) Plot the velocity-time graph of the motion. (b) Determine the distance the car travels during the first 2 seconds. (c) What distance does the car travel during the first 4 seconds? (d) What distance does the car travel during the entire 8 seconds? (e) Find the slope of the line between t = 0 and t = 4 seconds. What does this slope represent? (f) Find the slope of the line between t = 5 seconds and t = 7 seconds. What does this slope represent? 149 linear 9m 34 m 110 m 4 m/s2 slope = 1 m/s2 constant velocity An object is in free fall for five seconds. (a) Compute the total distance the object has fallen at the end of each second by using the proper kinematic formulas. (b) Use the distances calculated in part a to plot a positiontime graph. (c) Find the slope of the curve at the end of 2 seconds. (d) Find the slope of the curve at the end of 4 seconds. Answers (a) (b) (c) (d) 150 49 m parabola 19.6 m/s 39 m/s Use graphing to solve the following problem. Pressing the brake of a car caused it to slow down from 30.0 m/s to 20.0 m/s in 8.00 seconds. How far did the car travel during these 8.0 seconds? Answers 200 m Answers (a) (b) (c) (d) 75 m 150 m 125 m 500 m 151 Use the velocity-time the graph shown in the figure to determine how far an object moves during the first 4.5 seconds. Solve the problem both by counting rectangles and by using the formula for the area of a triangle. Answers 49.95 m 152 Determine the distance traveled by the runner whose velocitytime graph is given in the figure. Consider the full 8 second interval. 155 The speed of an airplane increased during a 5.0 s interval according to the data in the table. (a) Make a velocity-time graph of the motion (b) Find the distance (path length) traveled by the airplane during the first 3.0 seconds (c) Find the acceleration of the plane at the end of 2.0 seconds. (d) How does the acceleration obtained in part c compare with the acceleration at the end of 3.0 seconds? Answers 66 m 153 Time s Velocity m/s 0.0 1.0 2.0 3.0 4.0 5.0 30.0 40.0 50.0 60.0 70.0 80.0 Answers (a) (b) (c) (d) How far will a runner travel whose velocity time graph is shown in the figure? 156 linear 135 m 10 m/s2 10 m/s A race car traveling at +44 m/s is uniformly accelerated to a velocity of + 22 m/s over an 11-s interval. (a) Draw a velocity time graph of the motion of the car. (b) What is its displacement during this time? Answers (a) graph (b) 363 m Answers 157 35 m 154 The velocity is plotted against time for an object in the figure. How far does the object move in 6 seconds? A rocket traveling at +88 m/s is accelerated uniformly to +132 m/s over a 15 s interval. (a) Draw a velocity-time graph of the motion of the rocket (b) What is its displacement during this time? Answers (a) graph (b) 1650 m 158 A car accelerates at a constant rate from 15 m/s to 25 m/s while it travels 125 m. (a) Draw a velocity-time graph of the motion of the car (b) How long does this motion take? Answers (a) graph (b) 6.3 s Answers 90 m 159 The velocity of an automobile changes over an 8.0-s time period as shown in the table. (a) Plot the velocity-time graph of the motion. (b) Determine the displacement of the car during the first 2.0 s. (c) What displacement does the car have during the first 4.0 s? (d) What displacement does the car have during the entire 8.0 s? (e) Find the slope of the line between t = 0 s and t = 4.0 s. What does this slope represent? (f) Find the slope of the line between t = 5.0 s and t = 7.0 s. (g) What does this slope indicate? Time (s) 0.0 1.0 2.0 3.0 4.0 Velocity (m/s) 0.0 4.0 8.0 12.0 16.0 Time (s) 5.0 6.0 7.0 8.0 162 The velocity of an object over a 30 second time interval is shown in the figure. (a) Find the distance the moving object travels between t= 0 s and t= 5 s. (b) Find the distance the moving object travels between t = 5 s and t = 10 s. (c) Find the distance the moving object travels between t = 10 s and t = 15 s. (d) Find the distance the moving object travels between t = 0 s and t = 25 s. Velocity (m/s) 20.0 20.0 20.0 20.0 Answers (a) (b) (c) (d) (e) (f) (g) 160 8.0 m 32 m 110 m 4 m/s2 0 constant velocity A plane that is uniformly accelerated from 66 m/s to 88 m/s in 12 s. (a) Draw a velocity-time graph of the motion (b) How far did the plane travel? Answers (a) graph (b) 924 m/s 161 Use graphing to determine how far a plane flies in 15 s while its velocity is changing from 145 m/s to 75 m/s at a uniform rate of acceleration? Answers 1650 m Answers (a) (b) (c) (d) 163 75 m 150 m 125 m 500 m A plane flies in a straight line at a constant velocity of +75 m/s. assume that it is at the reference point when the clock reads t = 0 (a) Construct a table showing the position or displacement of the plane at the end of each second for a 10-s period. (b) Use the data from the table to plot a position-time graph . (c) Show that the slope of the line is the velocity of the plane. Use at least two different sets of points along the line. (d) Plot a velocity-time graph of the plane's motion for the first 6 s of the 10-s interval. (e) From the velocity-time graph, find the displacement of the plane between the second and the sixth period. Answers Clock Position (s) (m) 0 0 1 75 2 150 3 225 4 300 5 375 6 450 7 525 8 600 9 675 10 750 (b) (c) 75 m/s (d) (e) 300 m 164 Mary jogs for 15 min. at 240 m/min., walks the next 10 min. at 90 m/min., rests for 5 min., and jogs back to where she started at -180 m/min.( a) Plot a velocity-time graph for Mary's exercise run. (b) Find the area under the curve for the first 15 min. What does this represent? (c) What is the total distance traveled by Mary? (d) What is Mary's displacement from start to finish? 168 An object in free fall drops at a rate of -9.8 m/s2 (a) Make a table of the velocities of an object at the end of each second for the first 5 seconds of free-fall from rest. (b) Use the data in your table to plot a velocity-time graph. (c) What does the total area under the curve represent? (d) Calculate that value. Answers Answers (a) Time Velocity 0 0 1 -9.8 m/s 2 -19.6 m/s 3 -29.4 m/s 4 -39.2 m/s 5 -49.0 m/s (b) graph diagonal line (c) displacement (d) 122.5 meters (a) (b) 3600 m (c) 9000 m (d) 0 m 165 A plane flies in a straight line with a constant velocity of +5.0 x 101 m/s. (a) Construct a table showing the position or total displacement of the plane at the end of each second for a ten second period. (b) Use the data from the table to plot a position-time graph. (c) Show that the slope of the line on the position-time graph gives the velocity of the plane. Use at least two different sets of points along the graph. (d) Plot a velocity-time graph of the plane's motion for the first 6 seconds of the ten second interval. (e) Using the velocity-time graph find the displacement of the plane between the seventh and tenth seconds. 169 Answers Answers (a) (b) (c) (d) (e) (a) sketch graph (b) Area is 300 km (c) 75 km straight line about 50 m/s horizontal straight line 150 m 170 166 A car moves along a straight road at a constant velocity of 40 m/s south: (a) Plot its position-time graph for a ten-second interval. (b) Find the slope of the curve using two different points along the line (c) Plot a velocity-time graph for the car. What does the area under the curve of the graph represent? (d) Calculate the area under the curve of the graph between the fifth and sixth seconds. What does this area represent? Answers (a) a straight line (b) ∆y/∆x = 40 m/s (c) Horizontal straight line. The area under the curve is v x t and thus represents the total distance. (d) 40 m, the distance traveled during one second. 167 A car moves along a straight road at a constant velocity of +75 km/h for 4.0 h, stops for 2.0 h, and then drives in the reverse direction at the original speed for 3.0 h. (a) Plot a velocity-time graph for the car. (b) Find the area under the curve for the first 4 h. What does this represent? (c) Explain how to use the graph to find the distance the car is from its starting point at the end of 9.0 h. (d) Draw the position-time graph of the car's movement. Look at the figure. (a) What kind of motion does this graph represent? (b) What does the area under the curve of the graph represent? Answers (a) The graph represents motion with a positive increasing velocity. (b) The area under the curve represents the change in displacement. A person drives a car at a constant +25 m/s for 15.0 min. The car runs out of gas so the driver carrying an empty gas can walks at +1.5 m/s for 25 minutes to the nearest gas station. After 10 minutes needed to fill the can, the driver walks back to the car at a slower rate of -1.2 m/s. The car is then driven home at -20 m/s. (a) Draw a velocity time graph for the driver (use seconds as your time unit) (b) How long does it take the drive to walk back to the car. (c) At what time does he arrive back at the car (minutes) (d) How long does it take the driver to drive back home from where he filled the car. (e) At what time does he arrive home. (f) Draw a position-time graph from the areas under the curves of the velocity-time graph Answers Hard (a) (b) 31.25 min or 1,875 seconds (c) 81.25 minutes (d) 18.75 min or 1,125 s (e) 100 min (f) 171 The velocity-versus-time graph for a shuttle bus moving along a straight path is shown in the figure. Is the shuttle bus always moving in the same direction? Explain and refer to the time intervals shown on the graph. 173 As two trains move along a track, their conductors suddenly notice that they are headed toward each other. The figure gives their velocities v as functions of time t as the conductors slow the trains. The slowing processes begin when the trains are 200 m apart. What is their separation when both trains have stopped? Answers 40 m Answers No; the bus is moving in the positive direction for 30 s to 210 s (when velocity is positive) and in the negative direction from 275 s to 600 s (when the velocity is negative). 172 174 The figure shows a motion map and a position time graph of the car’s motion. Draw its velocity time graph The figure shows the position-time graph (cm) and the velocity-time graph (m/s) of a karate expert using a fist to break wooden boards during a 14 ms interval. (a) Use the velocity-time graph to describe the motion of the expert's fist during the first 10 ms. (b) Estimate the slope of the velocity-time graph to determine the acceleration of the fist when it suddenly stops. (c) Express the acceleration as a multiple of the gravitational acceleration, g = 9.80 m/s2. (d) Estimate the area under the velocity-time curve to find the displacement of the fist in the first 6 ms. Compare with the position-time graph. Answer 175 A man walks to the corner to mail a letter. Sketch two graphs showing his velocity and positions plotted against time. Answers Answers (a) The fist moves downward at about -13 m/s for about 5 m/s, then comes to a halt (b) 5.2 x 103 m/s2 (c) acceleration is about 530 g's (d) -8 cm; this agrees with the position-time graph, which shows a net displacement of -8 cm. 176 John rode his bicycle as fast as he could from his house through town and up Hemlock hill to Tom's house which is at the crest of the hill. He then rode back as fast as he could along the same route. Sketch a position-time graph of his motion. From this graph sketch a velocity-time graph. Answers 177 A velocity-time graph of a toy train is shown in the figure. Describe, in words, the velocity of the toy train between 0 and 40 seconds. 180 Use the intervals marked on the graph in the figure to describe the motion of the object. Figure 3 200 150 Position 100 50 0 O A B C D E F G H Time Answers Answers Starting from rest, it accelerates from rest to 10 m/s in the first 5 seconds. It remains at this speed for 10 s, before slowing to 4 m/s over the next 5 seconds. It eventually decelerates and comes to rest in the last 15 seconds. 178 181 The position time graph of a moving object is shown. Draw a velocity-time graph to accompany the position time graph shown. The figure shows a motion map and a position time graph of the car’s motion. Draw its velocity time graph Answers 182 The position time graph of a moving object is shown. Draw a velocity-time graph to accompany the position time graph shown. Answers 179 Sketch velocity-time graphs for the graphs shown in the figure. Answers 183 The position time graph of a moving object is shown. Draw a velocity-time graph to accompany the position time graph shown. Answers Answers 184 Two people leave a lamppost at the same time. One walks east, the other west, both at the same speed. (a) Describe the position-time of the two people (b) Sketch a position time graph of the two people (c) Describe velocity-time graphs of the two people. (d) Sketch the velocity time graph of the two people 188 Answers Call east the position direction and person A is the one who walks east. The position-time graph of person A is a straight line with a constant positive slope, Va. Person B's straight line has a constant negative slope of -Va. The velocity-time curve for person A is a horizontal line with a constant value of Va Person B is a horizontal line with a constant value of -Va. 185 Sketch the following graphs: (a) A position-time graph which always has a negative velocity. (b) A position-time graph which always has zero velocity. (c) A position-time graph which always has a positive velocity. Answers good summary problem. Answers 186 As a runaway scientific balloon ascends at 19.6 m/s, one of its instrument packages breaks free of a harness and free-falls. The figure gives the vertical velocity of the package versus time, from before it breaks free to when it reaches the ground. (a) Draw a picture of the package as it is released and fall (b) What is the acceleration of the package? (c) What is the velocity at 4 s, 5s, 6s, 7s and 8s? (d) What is displacement of the package as it rises and before it is released. (e) What maximum height.above the breakfree point does it rise? (f) How far does the package fall from its high point to the ground? (g) How high is the break-free point above the ground? (a) see picture (b) -9.8 m/s2 (c) 0 m/s, 9.8 m/s 19.6 m/s, 29.4 m/s, 39.2 m/s (d) 39.2 m (e) 19.6 m (f) 78.4 m g) 58.8 m Draw a position-time and an acceleration time graph for the motion of the particle shown on the velocity-time graph in the figure. 189 Use the velocity time graph shown. (a) Draw a position time graph of the object’s motion (b) Draw an acceleration time graph of the object’s motion. Make sure when you draw the graphs the time’s are aligned . Answers See answer book for graph 187 A car driving along a highway at a constant speed of 55 mph slows down to 25 mph as it enters a small village. In the center of town it is stopped by a traffic light. When the light changes he continues through the town. At the town boundary it speeds up to 65 mph and continues on his way. Sketch graphs, one above the other of the car's position, velocity, and acceleration, plotted against time. Answers Answers 100 m 190 The position time graph of a moving object is shown. (a) Draw a position-time graph to accompany the velocity time graph shown. (b) Draw an acceleration-time graph to accompany the position time graph shown. Time Answers 191 The velocity time graph of a moving object is shown. (a) Draw a position-time graph to accompany the velocity time graph shown. (b) Draw an acceleration-time graph to accompany the position time graph shown. Time Answers

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