Solution: Trip Home. You normally drive on the freeway

Does the speedometer of a car measure speed or velocity? Explain

The black dots at the top of Fig. Q2.2 represent a series of high-speed photographs of an insect flying in a straight line from left to right (in the positive x-direction). Which of the graphs in Fig. Q2.2 most plausibly depicts this insects motion?

Can an object with constant acceleration reverse its direction of travel? Can it reverse its direction twice? In both cases, explain your reasoning

Under what conditions is average velocity equal to instantaneous velocity?

Is it possible for an object to be (a) slowing down while its acceleration is increasing in magnitude; (b) speeding up while its acceleration is decreasing? In both cases, explain your reasoning.

Under what conditions does the magnitude of the average velocity equal the average speed?

When a Dodge Viper is at Elwoods Car Wash, a BMW Z3 is at Elm and Main. Later, when the Dodge reaches Elm and Main, the BMW reaches Elwoods Car Wash. How are the cars average velocities between these two times related?

A driver in Massachusetts was sent to traffic court for speeding. The evidence against the driver was that a policewoman observed the drivers car alongside a second car at a certain moment, and the policewoman had already clocked the second car going faster than the speed limit. The driver argued, The second car was passing me. I was not speeding. The judge ruled against the driver because, in the judges words, If two cars were side by side, both of you were speeding. If you were a lawyer representing the accused driver, how would you argue this case?

Can you have zero displacement and nonzero average velocity? Zero displacement and nonzero velocity? Illustrate your answers on an x-t graph.

Can you have zero acceleration and nonzero velocity? Use a vx@t graph to explain.

Can you have zero velocity and nonzero average acceleration? Zero velocity and nonzero acceleration? Use a vx@t graph to explain, and give an example of such motion.

An automobile is traveling west. Can it have a velocity toward the west and at the same time have an acceleration toward the east? Under what circumstances?

The officials truck in Fig. 2.2 is at x1 = 277 m at t1 = 16.0 s and is at x2 = 19 m at t2 = 25.0 s. (a) Sketch two different possible x-t graphs for the motion of the truck. (b) Does the average velocity vav@x during the time interval from t1 to t2 have the same value for both of your graphs? Why or why not?

Under constant acceleration the average velocity of a particle is half the sum of its initial and final velocities. Is this still true if the acceleration is not constant? Explain.

You throw a baseball straight up in the air so that it rises to a maximum height much greater than your height. Is the magnitude of the balls acceleration greater while it is being thrown or after it leaves your hand? Explain.

Prove these statements: (a) As long as you can ignore the effects of the air, if you throw anything vertically upward, it will have the same speed when it returns to the release point as when it was released. (b) The time of flight will be twice the time it takes to get to its highest point.

A dripping water faucet steadily releases drops 1.0 s apart. As these drops fall, does the distance between them increase, decrease, or remain the same? Prove your answer.

If you know the initial position and initial velocity of a vehicle and have a record of the acceleration at each instant, can you compute the vehicles position after a certain time? If so, explain how this might be done.

From the top of a tall building, you throw one ball straight up with speed v0 and one ball straight down with speed v0. (a) Which ball has the greater speed when it reaches the ground? (b) Which ball gets to the ground first? (c) Which ball has a greater displacement when it reaches the ground? (d) Which ball has traveled the greater distance when it hits the ground?

You run due east at a constant speed of 3.00 m/s for a distance of 120.0 m and then continue running east at a constant speed of 5.00 m/s for another 120.0 m. For the total 240.0-m run, is your average velocity 4.00 m/s, greater than 4.00 m/s, or less than 4.00 m/s? Explain.

An object is thrown straight up into the air and feels no air resistance. How can the object have an acceleration when it has stopped moving at its highest point?

When you drop an object from a certain height, it takes time T to reach the ground with no air resistance. If you dropped it from three times that height, how long (in terms of T) would it take to reach the ground?

A car travels in the +x-direction on a straight and level road. For the first 4.00 s of its motion, the average velocity of the car is vav@x = 6.25 m>s. How far does the car travel in 4.00 s?

In an experiment, a shearwater (a seabird) was taken from its nest, flown 5150 km away, and released. The bird found its way back to its nest 13.5 days after release. If we place the origin at the nest and extend the +x@axis to the release point, what was the birds average velocity in m>s (a) for the return flight and (b) for the whole episode, from leaving the nest to returning?

Trip Home. You normally drive on the freeway between San Diego and Los Angeles at an average speed of 105 km>h 165 mi>h2, and the trip takes 1 h and 50 min. On a Friday afternoon, however, heavy traffic slows you down and you drive the same distance at an average speed of only 70 km>h 143 mi>h2. How much longer does the trip take?

From Pillar to Post. Starting from a pillar, you run 200 m east (the +x@direction) at an average speed of 5.0 m>s and then run 280 m west at an average speed of 4.0 m>s to a post. Calculate (a) your average speed from pillar to post and (b) your average velocity from pillar to post

Starting from the front door of a ranch house, you walk 60.0 m due east to a windmill, turn around, and then slowly walk 40.0 m west to a bench, where you sit and watch the sunrise. It takes you 28.0 s to walk from the house to the windmill and then 36.0 s to walk from the windmill to the bench. For the entire trip from the front door to the bench, what are your (a) average velocity and (b) average speed?

A Honda Civic travels in a straight line along a road. The cars distance x from a stop sign is given as a function of time t by the equation x1t2 = at 2 - bt 3 , where a = 1.50 m>s 2 and b = 0.0500 m>s 3 . Calculate the average velocity of the car for each time interval: (a) t = 0 to t = 2.00 s; (b) t = 0 to t = 4.00 s; (c) t = 2.00 s to t = 4.00 s.

A car is stopped at a traffic light. It then travels along a straight road such that its distance from the light is given by x1t2 = bt2 - ct3 , where b = 2.40 m>s 2 and c = 0.120 m>s 3 . (a) Calculate the average velocity of the car for the time interval t = 0 to t = 10.0 s. (b) Calculate the instantaneous velocity of the car at t = 0, t = 5.0 s, and t = 10.0 s. (c) How long after starting from rest is the car again at rest?

A bird is flying due east. Its distance from a tall building is given by x1t2 = 28.0 m + 112.4 m>s2t - 10.0450 m>s 3 2t 3 . What is the instantaneous velocity of the bird when t = 8.00 s?

A ball moves in a straight line (the x-axis). The graph in Fig. E2.9 shows this ball’s velocity as a function of time. (a) What are the ball’s average speed and average velocity during the first 3.0 s? (b) Suppose that the ball moved in such a way that the graph segment after 2.0 s was -3.0 m/s instead of +3.0 m/s. Find the ball’s average speed and average velocity in this case.

A car travels in the \(+x \text {-direction }\) on a straight and level road. For the first 4.00 s of its motion, the average velocity of the car is \(v_{\mathrm{av}-\mathrm{x}}=6.25 \mathrm{\ m} / \mathrm{s}\). How far does the car travel in 4.00 s?

A test car travels in a straight line along the x-axis. The graph in Fig. E2.11 shows the cars position x as a function of time. Find its instantaneous velocity at points A through G.

Figure E2.12 shows the velocity of a solar-powered car as a function of time. The driver accelerates from a stop sign, cruises for 20 s at a constant speed of 60 km/h, and then brakes to come to a stop 40 s after leaving the stop sign. (a) Compute the average acceleration during these time intervals: (i) t = 0 to t = 10 s; (ii) t = 30 s to t = 40 s; (iii) t = 10 s to t = 30 s; (iv) t = 0 to t = 40 s. (b) What is the instantaneous acceleration at t = 20 s and at t = 35 s?

The Fastest (and Most Expensive) Car! The table shows test data for the Bugatti Veyron Super Sport, the fastest street car made. The car is moving in a straight line (the x-axis). (a) Sketch a vx@t graph of this cars velocity (in mi>h) as a function of time. Is its acceleration constant? (b) Calculate the cars average acceleration (in m>s 2 ) between (i) 0 and 2.1 s; (ii) 2.1 s and 20.0 s; (iii) 20.0 s and 53 s. Are these results consistent with your graph in part (a)? (Before you decide to buy this car, it might be helpful to know that only 300 will be built, it runs out of gas in 12 minutes at top speed, and it costs more than $1.5 million!)

A race car starts from rest and travels east along a straight and level track. For the first 5.0 s of the cars motion, the eastward component of the cars velocity is given by vx1t2 = 10.860 m>s 3 2t 2 . What is the acceleration of the car when vx = 12.0 m>s?

A turtle crawls along a straight line, which we will call the x-axis with the positive direction to the right. The equation for the turtle’s position as a function of time is \(x(t) = 50.0 \ \mathrm{cm} + 12.00 \ \mathrm{cm/s})t - (0.0625 \ \mathrm {cm/s}^2)t^2\). (a) Find the turtle’s initial velocity, initial position, and initial acceleration. (b) At what time t is the velocity of the turtle zero? (c) How long after starting does it take the turtle to return to its starting point? (d) At what times t is the turtle a distance of 10.0 cm from its starting point? What is the velocity (magnitude and direction) of the turtle at each of those times? (e) Sketch graphs of x versus \(t,v_x\) versus t, and \(a_x\) versus t, for the time interval t = 0 to t = 40 s.

An astronaut has left the International Space Station to test a new space scooter. Her partner measures the following velocity changes, each taking place in a 10-s interval. What are the magnitude, the algebraic sign, and the direction of the average acceleration in each interval? Assume that the positive direction is to the right. (a) At the beginning of the interval, the astronaut is moving toward the right along the x-axis at 15.0 m>s, and at the end of the interval she is moving toward the right at 5.0 m>s. (b) At the beginning she is moving toward the left at 5.0 m>s, and at the end she is moving toward the left at 15.0 m>s. (c) At the beginning she is moving toward the right at 15.0 m>s, and at the end she is moving toward the left at 15.0 m>s

A cars velocity as a function of time is given by vx1t2 = a + bt 2 , where a = 3.00 m>s and b = 0.100 m>s 3 . (a) Calculate the average acceleration for the time interval t = 0 to t = 5.00 s. (b) Calculate the instantaneous acceleration for t = 0 and t = 5.00 s. (c) Draw vx@t and ax@t graphs for the cars motion between t = 0 and t = 5.00 s.

The position of the front bumper of a test car under microprocessor control is given by \(x(t) = 2.17 \ \mathrm m + (4.80 \ \mathrm {m/s}^2)t^2 - (0.100 \ \mathrm {m/s}^6)t^6\) . (a) Find its position and acceleration at the instants when the car has zero velocity. (b) Draw \(x-t, v_{x}-t\), and \(a_{x}-t\) graphs for the motion of the bumper between t = 0 and t = 2.00 s.

An antelope moving with constant acceleration covers the distance between two points 70.0 m apart in 6.00 s. Its speed as it passes the second point is 15.0 m>s. What are (a) its speed at the first point and (b) its acceleration?

Blackout? A jet fighter pilot wishes to accelerate from rest at a constant acceleration of 5g to reach Mach 3 (three times the speed of sound) as quickly as possible. Experimental tests reveal that he will black out if this acceleration lasts for more than 5.0 s. Use 331 m/s for the speed of sound. (a) Will the period of acceleration last long enough to cause him to black out? (b) What is the greatest speed he can reach with an acceleration of 5g before he blacks out?

. A Fast Pitch. The fastest measured pitched baseball left the pitchers hand at a speed of 45.0 m>s. If the pitcher was in contact with the ball over a distance of 1.50 m and produced constant acceleration, (a) what acceleration did he give the ball, and (b) how much time did it take him to pitch it?

A Tennis Serve. In the fastest measured tennis serve, the ball left the racquet at 73.14 m>s. A served tennis ball is typically in contact with the racquet for 30.0 ms and starts from rest. Assume constant acceleration. (a) What was the balls acceleration during this serve? (b) How far did the ball travel during the serve?

Automobile Air Bags. The human body can survive an acceleration trauma incident (sudden stop) if the magnitude of the acceleration is less than 250 m>s 2 . If you are in an automobile accident with an initial speed of 105 km>h 165 mi>h2 and are stopped by an airbag that inflates from the dashboard, over what distance must the airbag stop you for you to survive the crash?

A pilot who accelerates at more than 4g begins to gray out but doesnt completely lose consciousness. (a) Assuming constant acceleration, what is the shortest time that a jet pilot starting from rest can take to reach Mach 4 (four times the speed of sound) without graying out? (b) How far would the plane travel during this period of acceleration? (Use 331 m>s for the speed of sound in cold air.)

Air-Bag Injuries. During an auto accident, the vehicles air bags deploy and slow down the passengers more gently than if they had hit the windshield or steering wheel. According to safety standards, air bags produce a maximum acceleration of 60g that lasts for only 36 ms (or less). How far (in meters) does a person travel in coming to a complete stop in 36 ms at a constant acceleration of 60g?

Prevention of Hip Fractures. Falls resulting in hip fractures are a major cause of injury and even death to the elderly. Typically, the hips speed at impact is about 2.0 m>s. If this can be reduced to 1.3 m>s or less, the hip will usually not fracture. One way to do this is by wearing elastic hip pads. (a) If a typical pad is 5.0 cm thick and compresses by 2.0 cm during the impact of a fall, what constant acceleration (in m>s 2 and in gs) does the hip undergo to reduce its speed from 2.0 m>s to 1.3 m>s? (b) The acceleration you found in part (a) may seem rather large, but to assess its effects on the hip, calculate how long it lasts.

Are We Martians? It has been suggested, and not facetiously, that life might have originated on Mars and been carried to the earth when a meteor hit Mars and blasted pieces of rock (perhaps containing primitive life) free of the Martian surface. Astronomers know that many Martian rocks have come to the earth this way. (For instance, search the Internet for ALH 84001.) One objection to this idea is that microbes would have had to undergo an enormous lethal acceleration during the impact. Let us investigate how large such an acceleration might be. To escape Mars, rock fragments would have to reach its escape velocity of 5.0 km>s, and that would most likely happen over a distance of about 4.0 m during the meteor impact. (a) What would be the acceleration (in m>s 2 and gs) of such a rock fragment, if the acceleration is constant? (b) How long would this acceleration last? (c) In tests, scientists have found that over 40% of Bacillus subtilis bacteria survived after an acceleration of 450,000g. In light of your answer to part (a), can we rule out the hypothesis that life might have been blasted from Mars to the earth?

Entering the Freeway. A car sits on an entrance ramp to a freeway, waiting for a break in the traffic. Then the driver accelerates with constant acceleration along the ramp and onto the freeway. The car starts from rest, moves in a straight line, and has a speed of 20 m/s (45 mi/h) when it reaches the end of the 120-m- long ramp. (a) What is the acceleration of the car? (b) How much time does it take the car to travel the length of the ramp? (c) The traffic on the freeway is moving at a constant speed of 20 m/s. What distance does the traffic travel while the car is moving the length of the ramp?

At launch a rocket ship weighs 4.5 million pounds. When it is launched from rest, it takes 8.00 s to reach 161 km/h; at the end of the first 1.00 min, its speed is 1610 km/h. (a) What is the average acceleration \(\left(\text { in } \mathrm{m} / \mathrm{s}^{2}\right)\) of the rocket (i) during the first 8.00 s and (ii) between 8.00 s and the end of the first 1.00 min? (b) Assuming the acceleration is constant during each time interval (but not necessarily the same in both intervals), what distance does the rocket travel (i) during the first 8.00 s and (ii) during the interval from 8.00 s to 1.00 min?

A cat walks in a straight line, which we shall call the x-axis, with the positive direction to the right. As an observant physicist, you make measurements of this cats motion and construct a graph of the felines velocity as a function of time (Fig. E2.30). (a) Find the cats velocity at t = 4.0 s and at t = 7.0 s. (b) What is the cats acceleration at t = 3.0 s? At t = 6.0 s? At t = 7.0 s? (c) What distance does the cat move during the first 4.5 s? From t = 0 to t = 7.5 s? (d) Assuming that the cat started at the origin, sketch clear graphs of the cats acceleration and position as functions of time.

The graph in Fig. E2.31 shows the velocity of a motorcycle police officer plotted as a function of time. (a) Find the instantaneous acceleration at t = 3 s, t = 7 s, and t = 11 s. (b) How far does the officer go in the first 5 s? The first 9 s? The first 13 s?

Two cars, A and B, move along the x-axis. Figure E2.32 is a graph of the positions of A and B versus time. (a) In motion diagrams (like Figs. 2.13b and 2.14b), show the position, velocity, and acceleration of each of the two cars at t = 0, t = 1 s, and t = 3 s. (b) At what time(s), if any, do A and B have the same position? (c) Graph velocity versus time for both A and B. (d) At what time(s), if any, do A and B have the same velocity? (e) At what time(s), if any, does car A pass car B? (f) At what time(s), if any, does car B pass car A?

A small block has constant acceleration as it slides down a frictionless incline. The block is released from rest at the top of the incline, and its speed after it has traveled 6.80 m to the bottom of the incline is 3.80 m>s. What is the speed of the block when it is 3.40 m from the top of the incline?

At the instant the traffic light turns green, a car that has been waiting at an intersection starts ahead with a constant acceleration of 2.80 m>s 2 . At the same instant a truck, traveling with a constant speed of 20.0 m>s, overtakes and passes the car. (a) How far beyond its starting point does the car overtake the truck? (b) How fast is the car traveling when it overtakes the truck? (c) Sketch an x-t graph of the motion of both vehicles. Take x = 0 at the intersection. (d) Sketch a vx@t graph of the motion of both vehicles

(a) If a flea can jump straight up to a height of 0.440 m, what is its initial speed as it leaves the ground? (b) How long is it in the air?

A small rock is thrown vertically upward with a speed of 22.0 m/s from the edge of the roof of a 30.0-m-tall building. The rock doesn’t hit the building on its way back down and lands on the street below. Ignore air resistance. (a) What is the speed of the rock just before it hits the street? (b) How much time elapses from when the rock is thrown until it hits the street?

A juggler throws a bowling pin straight up with an initial speed of 8.20 m/s. How much time elapses until the bowling pin returns to the juggler’s hand?

You throw a glob of putty straight up toward the ceiling, which is 3.60 m above the point where the putty leaves your hand. The initial speed of the putty as it leaves your hand is 9.50 m>s. (a) What is the speed of the putty just before it strikes the ceiling? (b) How much time from when it leaves your hand does it take the putty to reach the ceiling?

A tennis ball on Mars, where the acceleration due to gravity is 0.379g and air resistance is negligible, is hit directly upward and returns to the same level 8.5 s later. (a) How high above its original point did the ball go? (b) How fast was it moving just after it was hit? (c) Sketch graphs of the ball’s vertical position, vertical velocity, and vertical acceleration as functions of time while it’s in the Martian air.

From Pillar to Post. Starting from a pillar, you run 200 m east \(\text { (the }+x \text {-direction) }\) at an average speed of 5.0 m/s and then run 280 m west at an average speed of 4.0 m/s to a post. Calculate (a) your average speed from pillar to post and (b) your average velocity from pillar to post.

A Simple Reaction-Time Test. A meter stick is held vertically above your hand, with the lower end between your thumb and first finger. When you see the meter stick released, you grab it with those two fingers. You can calculate your reaction time from the distance the meter stick falls, read directly from the point where your fingers grabbed it. (a) Derive a relationship for your reaction time in terms of this measured distance, d. (b) If the measured distance is 17.6 cm, what is your reaction time?

A brick is dropped (zero initial speed) from the roof of a building. The brick strikes the ground in 1.90 s. You may ignore air resistance, so the brick is in free fall. (a) How tall, in meters, is the building? (b) What is the magnitude of the bricks velocity just before it reaches the ground? (c) Sketch ay@t, vy@t, and y-t graphs for the motion of the brick

Launch Failure. A 7500-kg rocket blasts off vertically from the launch pad with a constant upward acceleration of 2.25 m>s 2 and feels no appreciable air resistance. When it has reached a height of 525 m, its engines suddenly fail; the only force acting on it is now gravity. (a) What is the maximum height this rocket will reach above the launch pad? (b) How much time will elapse after engine failure before the rocket comes crashing down to the launch pad, and how fast will it be moving just before it crashes? (c) Sketch ay@t, vy@t, and y-t graphs of the rockets motion from the instant of blast-off to the instant just before it strikes the launch pad.

A hot-air balloonist, rising vertically with a constant velocity of magnitude 5.00 m/s, releases a sandbag at an instant when the balloon is 40.0 m above the ground (Fig. E2.44). After the sandbag is released, it is in free fall. (a) Compute the position and velocity of the sandbag at 0.250 s and 1.00 s after its release. (b) How many seconds after its release does the bag strike the ground? (c) With what magnitude of velocity does it strike the ground? (d) What is the greatest height above the ground that the sandbag reaches? (e) Sketch \(a_y-t, v_y-t\), and y-t graphs for the motion.

The rocket-driven sled Sonic Wind No. 2, used for investigating the physiological effects of large accelerations, runs on a straight, level track 1070 m (3500 ft) long. Starting from rest, it can reach a speed of 224 m>s1500 mi>h2 in 0.900 s. (a) Compute the acceleration in m>s 2 , assuming that it is constant. (b) What is the ratio of this acceleration to that of a freely falling body (g)? (c) What distance is covered in 0.900 s? (d) A magazine article states that at the end of a certain run, the speed of the sled decreased from 283 m>s 1632 mi>h2 to zero in 1.40 s and that during this time the magnitude of the acceleration was greater than 40g. Are these figures consistent?

An egg is thrown nearly vertically upward from a point near the cornice of a tall building. The egg just misses the cornice on the way down and passes a point 30.0 m below its starting point 5.00 s after it leaves the throwers hand. Ignore air resistance. (a) What is the initial speed of the egg? (b) How high does it rise above its starting point? (c) What is the magnitude of its velocity at the highest point? (d) What are the magnitude and direction of its acceleration at the highest point? (e) Sketch ay@t, vy@t, and y-t graphs for the motion of the egg.

A 15-kg rock is dropped from rest on the earth and reaches the ground in 1.75 s. When it is dropped from the same height on Saturns satellite Enceladus, the rock reaches the ground in 18.6 s. What is the acceleration due to gravity on Enceladus?

A large boulder is ejected vertically upward from a volcano with an initial speed of 40.0 m>s. Ignore air resistance. (a) At what time after being ejected is the boulder moving at 20.0 m>s upward? (b) At what time is it moving at 20.0 m>s downward? (c) When is the displacement of the boulder from its initial position zero? (d) When is the velocity of the boulder zero? (e) What are the magnitude and direction of the acceleration while the boulder is (i) moving upward? (ii) Moving downward? (iii) At the highest point? (f) Sketch ay@t, vy@t, and y-t graphs for the motion

You throw a small rock straight up from the edge of a highway bridge that crosses a river. The rock passes you on its way down, 6.00 s after it was thrown. What is the speed of the rock just before it reaches the water 28.0 m below the point where the rock left your hand? Ignore air resistance

A small object moves along the x-axis with acceleration \(a_x(t) = -(0.0320 \ \mathrm {m/s}^3 )(15.0 \ s - t)\). At t = 0 the object is at x = -14.0 m and has velocity \(v_{0x} = 8.00 \ \mathrm {m/s}\). What is the x-coordinate of the object when t = 10.0 s?

A rocket starts from rest and moves upward from the surface of the earth. For the first 10.0 s of its motion, the vertical acceleration of the rocket is given by ay = 12.80 m>s 3 2t, where the +y-direction is upward. (a) What is the height of the rocket above the surface of the earth at t = 10.0 s? (b) What is the speed of the rocket when it is 325 m above the surface of the earth?

The acceleration of a bus is given by \(a_x(t)=\alpha t\text{ where }\alpha=1.2\mathrm{\ m}/\mathrm{s}^3\text{. }\) (a) If the bus’s velocity at time t = 1.0 s is 5.0 m/s, what is its velocity at time t = 2.0 s? (b) If the bus’s position at time t = 1.0 s is 6.0 m, what is its position at time t = 2.0 s? (c) Sketch \(a_y-t,\ v_y-t,\text{ and }x-t\) graphs for the motion.

The acceleration of a motorcycle is given by ax1t2 = At - Bt2 , where A = 1.50 m>s 3 and B = 0.120 m>s 4 . The motorcycle is at rest at the origin at time t = 0. (a) Find its position and velocity as functions of time. (b) Calculate the maximum velocity it attains.

Flying Leap of the Flea. High-speed motion pictures 13500 frames>second2 of a jumping, 210@mg flea yielded the data used to plot the graph in Fig. E2.54. (See The Flying Leap of the Flea by M. Rothschild, Y. Schlein, K. Parker, C. Neville, and S. Sternberg in the November 1973 Scientific American.) This flea was about 2 mm long and jumped at a nearly vertical takeoff angle. Use the graph to answer these questions: (a) Is the acceleration of the flea ever zero? If so, when? Justify your answer. (b) Find the maximum height the flea reached in the first 2.5 ms. (c) Find the fleas acceleration at 0.5 ms, 1.0 ms, and 1.5 ms. (d) Find the fleas height at 0.5 ms, 1.0 ms, and 1.5 ms.

A typical male sprinter can maintain his maximum acceleration for 2.0 s, and his maximum speed is 10 m>s. After he reaches this maximum speed, his acceleration becomes zero, and then he runs at constant speed. Assume that his acceleration is constant during the first 2.0 s of the race, that he starts from rest, and that he runs in a straight line. (a) How far has the sprinter run when he reaches his maximum speed? (b) What is the magnitude of his average velocity for a race of these lengths: (i) 50.0 m; (ii) 100.0 m; (iii) 200.0 m?

A lunar lander is descending toward the moons surface. Until the lander reaches the surface, its height above the surface of the moon is given by y1t2 = b - ct + dt2 , where b = 800 m is the initial height of the lander above the surface, c = 60.0 m>s, and d = 1.05 m>s 2 . (a) What is the initial velocity of the lander, at t = 0? (b) What is the velocity of the lander just before it reaches the lunar surface?

Earthquake Analysis. Earthquakes produce several types of shock waves. The most well known are the P-waves (P for primary or pressure) and the S-waves (S for secondary or shear). In the earths crust, P-waves travel at about 6.5 km>s and S-waves move at about 3.5 km>s. The time delay between the arrival of these two waves at a seismic recording station tells geologists how far away an earthquake occurred. If the time delay is 33 s, how far from the seismic station did the earthquake occur?

A brick is dropped from the roof of a tall building. After it has been falling for a few seconds, it falls 40.0 m in a 1.00-s time interval. What distance will it fall during the next 1.00 s? Ignore air resistance

A rocket carrying a satellite is accelerating straight up from the earths surface. At 1.15 s after liftoff, the rocket clears the top of its launch platform, 63 m above the ground. After an additional 4.75 s, it is 1.00 km above the ground. Calculate the magnitude of the average velocity of the rocket for (a) the 4.75-s part of its flight and (b) the first 5.90 s of its flight.

A subway train starts from rest at a station and accelerates at a rate of 1.60 m>s 2 for 14.0 s. It runs at constant speed for 70.0 s and slows down at a rate of 3.50 m>s 2 until it stops at the next station. Find the total distance covered.

A gazelle is running in a straight line (the x-axis). The graph in Fig. P2.61 shows this animals velocity as a function of time. During the first 12.0 s, find (a) the total distance moved and (b) the displacement of the gazelle. (c) Sketch an ax@t graph showing this gazelles acceleration as a function of time for the first 12.0 s.

Collision. The engineer of a passenger train traveling at 25.0 m>s sights a freight train whose caboose is 200 m ahead on the same track (Fig. P2.62). The freight train is traveling at 15.0 m>s in the same direction as the passenger train. The engineer of the passenger train immediately applies the brakes, causing a constant acceleration of 0.100 m>s2 in a direction opposite to the trains velocity, while the freight train continues with constant speed. Take x = 0 at the location of the front of the passenger train when the engineer applies the brakes. (a) Will the cows nearby witness a collision? (b) If so, where will it take place? (c) On a single graph, sketch the positions of the front of the passenger train and the back of the freight train.

A ball starts from rest and rolls down a hill with uniform acceleration, traveling 200 m during the second 5.0 s of its motion. How far did it roll during the first 5.0 s of motion?

Two cars start 200 m apart and drive toward each other at a steady 10 m>s. On the front of one of them, an energetic grasshopper jumps back and forth between the cars (he has strong legs!) with a constant horizontal velocity of 15 m>s relative to the ground. The insect jumps the instant he lands, so he spends no time resting on either car. What total distance does the grasshopper travel before the cars hit?

A car and a truck start from rest at the same instant, with the car initially at some distance behind the truck. The truck has a constant acceleration of \(2.10 \ \mathrm {m/s}^2\), and the car has an acceleration of \(3.40 \ \mathrm{m/s}^2\). The car overtakes the truck after the truck has moved 60.0 m. (a) How much time does it take the car to overtake the truck? (b) How far was the car behind the truck initially? (c) What is the speed of each when they are abreast? (d) On a single graph, sketch the position of each vehicle as a function of time. Take x = 0 at the initial location of the truck.

You are standing at rest at a bus stop. A bus moving at a constant speed of 5.00 m>s passes you. When the rear of the bus is 12.0 m past you, you realize that it is your bus, so you start to run toward it with a constant acceleration of 0.960 m>s 2 . How far would you have to run before you catch up with the rear of the bus, and how fast must you be running then? Would an average college student be physically able to accomplish this?

Passing. The driver of a car wishes to pass a truck that is traveling at a constant speed of 20.0 m/s (about 45 mi/h). Initially, the car is also traveling at 20.0 m/s, and its front bumper is 24.0 m behind the truck’s rear bumper. The car accelerates at a constant \(0.600 \mathrm{\ m} / \mathrm{s}^{2}\), then pulls back into the truck’s lane when the rear of the car is 26.0 m ahead of the front of the truck. The car is 4.5 m long, and the truck is 21.0 m long. (a) How much time is required for the car to pass the truck? (b) What distance does the car travel during this time? (c) What is the final speed of the car?

An objects velocity is measured to be vx1t2 = a - bt 2 , where a = 4.00 m>s and b = 2.00 m>s 3 . At t = 0 the object is at x = 0. (a) Calculate the objects position and acceleration as functions of time. (b) What is the objects maximum positive displacement from the origin?

The acceleration of a particle is given by ax1t2 = -2.00 m>s 2 + 13.00 m>s 3 2t. (a) Find the initial velocity v0x such that the particle will have the same x-coordinate at t = 4.00 s as it had at t = 0. (b) What will be the velocity at t = 4.00 s?

Egg Drop. You are on the roof of the physics building, 46.0 m above the ground (Fig. P2.70). Your physics professor, who is 1.80 m tall, is walking alongside the building at a constant speed of 1.20 m>s. If you wish to drop an egg on your professors head, where should the professor be when you release the egg? Assume that the egg is in free fall.

A certain volcano on earth can eject rocks vertically to a maximum height H. (a) How high (in terms of H) would these rocks go if a volcano on Mars ejected them with the same initial velocity? The acceleration due to gravity on Mars is \(3.71 \mathrm{\ m} / \mathrm{s}^{2}\) ; ignore air resistance on both planets. (b) If the rocks are in the air for a time T on earth, for how long (in terms of T) would they be in the air on Mars?

An entertainer juggles balls while doing other activities. In one act, she throws a ball vertically upward, and while it is in the air, she runs to and from a table 5.50 m away at an average speed of 3.00 m>s, returning just in time to catch the falling ball. (a) With what minimum initial speed must she throw the ball upward to accomplish this feat? (b) How high above its initial position is the ball just as she reaches the table?

Look Out Below. Sam heaves a 16-lb shot straight up, giving it a constant upward acceleration from rest of 35.0 m>s 2 for 64.0 cm. He releases it 2.20 m above the ground. Ignore air resistance. (a) What is the speed of the shot when Sam releases it? (b) How high above the ground does it go? (c) How much time does he have to get out of its way before it returns to the height of the top of his head, 1.83 m above the ground?

A flowerpot falls off a windowsill and passes the window of the story below. Ignore air resistance. It takes the pot 0.380 s to pass from the top to the bottom of this window, which is 1.90 m high. How far is the top of the window below the windowsill from which the flowerpot fell?

Two stones are thrown vertically upward from the ground, one with three times the initial speed of the other. (a) If the faster stone takes 10 s to return to the ground, how long will it take the slower stone to return? (b) If the slower stone reaches a maximum height of H, how high (in terms of H) will the faster stone go? Assume free fall.

A Multistage Rocket. In the first stage of a two-stage rocket, the rocket is fired from the launch pad starting from rest but with a constant acceleration of 3.50 m>s 2 upward. At 25.0 s after launch, the second stage fires for 10.0 s, which boosts the rockets velocity to 132.5 m>s upward at 35.0 s after launch. This firing uses up all of the fuel, however, so after the second stage has finished firing, the only force acting on the rocket is gravity. Ignore air resistance. (a) Find the maximum height that the stagetwo rocket reaches above the launch pad. (b) How much time after the end of the stage-two firing will it take for the rocket to fall back to the launch pad? (c) How fast will the stage-two rocket be moving just as it reaches the launch pad?

During your summer internship for an aerospace company, you are asked to design a small research rocket. The rocket is to be launched from rest from the earths surface and is to reach a maximum height of 960 m above the earths surface. The rockets engines give the rocket an upward acceleration of 16.0 m>s 2 during the time T that they fire. After the engines shut off, the rocket is in free fall. Ignore air resistance. What must be the value of T in order for the rocket to reach the required altitude?

A physics teacher performing an outdoor demonstration suddenly falls from rest off a high cliff and simultaneously shouts “Help.” When she has fallen for 3.0 s, she hears the echo of her shout from the valley floor below. The speed of sound is 340 m/s. (a) How tall is the cliff? (b) If we ignore air resistance, how fast will she be moving just before she hits the ground? (Her actual speed will be less than this, due to air resistance.)

A helicopter carrying Dr. Evil takes off with a constant upward acceleration of 5.0 m>s 2 . Secret agent Austin Powers jumps on just as the helicopter lifts off the ground. After the two men struggle for 10.0 s, Powers shuts off the engine and steps out of the helicopter. Assume that the helicopter is in free fall after its engine is shut off, and ignore the effects of air resistance. (a) What is the maximum height above ground reached by the helicopter? (b) Powers deploys a jet pack strapped on his back 7.0 s after leaving the helicopter, and then he has a constant downward acceleration with magnitude 2.0 m>s 2 . How far is Powers above the ground when the helicopter crashes into the ground?

Cliff Height. You are climbing in the High Sierra when you suddenly find yourself at the edge of a fog-shrouded cliff. To find the height of this cliff, you drop a rock from the top; 8.00 s later you hear the sound of the rock hitting the ground at the foot of the cliff. (a) If you ignore air resistance, how high is the cliff if the speed of sound is 330 m/s? (b) Suppose you had ignored the time it takes the sound to reach you. In that case, would you have overestimated or underestimated the height of the cliff? Explain.

An object is moving along the x-axis. At t = 0 it has velocity v0x = 20.0 m>s. Starting at time t = 0 it has acceleration ax = -Ct, where C has units of m>s 3 . (a) What is the value of C if the object stops in 8.00 s after t = 0? (b) For the value of C calculated in part (a), how far does the object travel during the 8.00 s?

A ball is thrown straight up from the ground with speed v0. At the same instant, a second ball is dropped from rest from a height H, directly above the point where the first ball was thrown upward. There is no air resistance. (a) Find the time at which the two balls collide. (b) Find the value of H in terms of v0 and g such that at the instant when the balls collide, the first ball is at the highest point of its motion.

Cars A and B travel in a straight line. The distance of A from the starting point is given as a function of time by xA1t2 = at + bt 2 , with a = 2.60 m>s and b = 1.20 m>s 2 . The distance of B from the starting point is xB1t2 = gt 2 - dt 3 , with g = 2.80 m>s 2 and d = 0.20 m>s 3 . (a) Which car is ahead just after the two cars leave the starting point? (b) At what time(s) are the cars at the same point? (c) At what time(s) is the distance from A to B neither increasing nor decreasing? (d) At what time(s) do A and B have the same acceleration?

In your physics lab you release a small glider from rest at various points on a long, frictionless air track that is inclined at an angle u above the horizontal. With an electronic photocell, you measure the time t it takes the glider to slide a distance x from the release point to the bottom of the track. Your measurements are given in Fig. P2.84, which shows a second-order polynomial (quadratic) fit to the plotted data. You are asked to find the gliders acceleration, which is assumed to be constant. There is some error in each measurement, so instead of using a single set of x and t values, you can be more accurate if you use graphical methods and obtain your measured value of the acceleration from the graph. (a) How can you re-graph the data so that the data points fall close to a straight line? (Hint: You might want to plot x or t, or both, raised to some power.) (b) Construct the graph you described in part (a) and find the equation for the straight line that is the best fit to the data points. (c) Use the straightline fit from part (b) to calculate the acceleration of the glider. (d) The glider is released at a distance x = 1.35 m from the bottom of the track. Use the acceleration value you obtained in part (c) to calculate the speed of the glider when it reaches the bottom of the track.

In a physics lab experiment, you release a small steel ball at various heights above the ground and measure the balls speed just before it strikes the ground. You plot your data on a graph that has the release height (in meters) on the vertical axis and the square of the final speed (in m2>s 2 ) on the horizontal axis. In this graph your data points lie close to a straight line. (a) Using g = 9.80 m>s 2 and ignoring the effect of air resistance, what is the numerical value of the slope of this straight line? (Include the correct units.) The presence of air resistance reduces the magnitude of the downward acceleration, and the effect of air resistance increases as the speed of the object increases. You repeat the experiment, but this time with a tennis ball as the object being dropped. Air resistance now has a noticeable effect on the data. (b) Is the final speed for a given release height higher than, lower than, or the same as when you ignored air resistance? (c) Is the graph of the release height versus the square of the final speed still a straight line? Sketch the qualitative shape of the graph when air resistance is present.

A model car starts from rest and travels in a straight line. A smartphone mounted on the car has an app that transmits the magnitude of the cars acceleration (measured by an accelerometer) every second. The results are given in the table: Each measured value has some experimental error. (a) Plot acceleration versus time and find the equation for the straight line that gives the best fit to the data. (b) Use the equation for a1t2 that you found in part (a) to calculate v1t2, the speed of the car as a function of time. Sketch the graph of v versus t. Is this graph a straight line? (c) Use your result from part (b) to calculate the speed of the car at t = 5.00 s. (d) Calculate the distance the car travels between t = 0 and t = 5.00 s.

In the vertical jump, an athlete starts from a crouch and jumps upward as high as possible. Even the best athletes spend little more than 1.00 s in the air (their hang time). Treat the athlete as a particle and let ymax be his maximum height above the floor. To explain why he seems to hang in the air, calculate the ratio of the time he is above ymax>2 to the time it takes him to go from the floor to that height. Ignore air resistance

Catching the Bus. A student is running at her top speed of 5.0 m>s to catch a bus, which is stopped at the bus stop. When the student is still 40.0 m from the bus, it starts to pull away, moving with a constant acceleration of 0.170 m>s 2 . (a) For how much time and what distance does the student have to run at 5.0 m>s before she overtakes the bus? (b) When she reaches the bus, how fast is the bus traveling? (c) Sketch an x-t graph for both the student and the bus. Take x = 0 at the initial position of the student. (d) The equations you used in part (a) to find the time have a second solution, corresponding to a later time for which the student and bus are again at the same place if they continue their specified motions. Explain the significance of this second solution. How fast is the bus traveling at this point? (e) If the students top speed is 3.5 m>s, will she catch the bus? (f) What is the minimum speed the student must have to just catch up with the bus? For what time and what distance does she have to run in that case?

A ball is thrown straight up from the edge of the roof of a building. A second ball is dropped from the roof 1.00 s later. Ignore air resistance. (a) If the height of the building is 20.0 m, what must the initial speed of the first ball be if both are to hit the ground at the same time? On the same graph, sketch the positions of both balls as a function of time, measured from when the first ball is thrown. Consider the same situation, but now let the initial speed v0 of the first ball be given and treat the height h of the building as an unknown. (b) What must the height of the building be for both balls to reach the ground at the same time if (i) v0 is 6.0 m>s and (ii) v0 is 9.5 m>s? (c) If v0 is greater than some value vmax, no value of h exists that allows both balls to hit the ground at the same time. Solve for vmax. The value vmax has a simple physical interpretation. What is it? (d) If v0 is less than some value vmin, no value of h exists that allows both balls to hit the ground at the same time. Solve for vmin. The value vmin also has a simple physical interpretation. What is it?

If the contraction of the left ventricle lasts 250 ms and the speed of blood flow in the aorta (the large artery leaving the heart) is 0.80 m>s at the end of the contraction, what is the average acceleration of a red blood cell as it leaves the heart? (a) 310 m>s 2 ; (b) 31 m>s 2 ; (c) 3.2 m>s 2 ; (d) 0.32 m>s 2 .

If the aorta (diameter da) branches into two equal-sized arteries with a combined area equal to that of the aorta, what is the diameter of one of the branches? (a) 1da; (b) da >12; (c) 2da; (d) da >2.

The velocity of blood in the aorta can be measured directly with ultrasound techniques. A typical graph of blood velocity versus time during a single heartbeat is shown in Fig. P2.92. Which statement is the best interpretation of this graph? (a) The blood flow changes direction at about 0.25 s; (b) the speed of the blood flow begins to decrease at about 0.10 s; (c) the acceleration of the blood is greatest in magnitude at about 0.25 s; (d) the acceleration of the blood is greatest in magnitude at about 0.10 s.