For what kind of motion are the instantaneous and average

Problem 1CQ You and your dog go for a walk to a nearby park. On the way, your dog takes many short side trips to chase squirrels, examine fire hydrants, and so on. When you arrive at the park, do you and your dog have the same displacement? Have you traveled the same distance? Explain.

Referring to Figure 2–20, you walk from your home to the library, then to the park. (a) What is the distance traveled? (b) What is your displacement?

Problem 2CQ Does an odometer in a car measure distance or displacement? Explain.

The two tennis players shown in Figure 2–21 walk to the net to congratulate one another. (a) Find the distance traveled and the displacement of player A. (b) Repeat for player B.

Can you drive your car in such a way that the distance it covers is (a) greater than, (b) equal to, or (c) less than the magnitude of its displacement? In each case, give an example if your answer is yes, explain why not if your answer is no.

Referring to Example 2–12 Suppose the balloon is descending with a constant speed of 4.2 m/s when the bag of sand comes loose at a height of 35 m. (a) How long is the bag in the air? (b) What is the speed of the bag when it is 15 m above the ground?

The golfer in Figure 2–22 sinks the ball in two putts, as shown. What are (a) the distance traveled by the ball, and (b) the displacement of the ball?

Problem 4CQ Art astronaut orbits Earth in the space shuttle. In one complete orbit, is the magnitude of the displacement the same as the distance traveled? Explain.

Problem 4P In Figure 2-20, you walk from the park to your friend's house, then back to your house. What is your (a) distance traveled, and (b) displacement?

Problem 5CQ After a tennis match the players dash to the net to congratulate one another. If they both run with a speed of 3 m/s, are their velocities equal? Explain.

A jogger runs on the track shown in Figure 2–23. Neglecting the curvature of the corners, (a) what is the distance traveled and the displacement in running from point A to point B? (b) Find the distance and displacement for a complete circuit of the track.

Problem 6CQ Does a speedometer measure speed or velocity? Explain.

IP A child rides a pony on a circular track whose radius is 4.5 m. (a) Find the distance traveled and the displacement after the child has gone halfway around the track, (b) Does the distance traveled increase, decrease, or stay the same when the child completes one circuit of the track? Explain, (c) Does the displacement increase, decrease, or stay the same when the child completes one circuit of the track? Explain, (d) Find the distance and displacement after a complete circuit of the track.

Problem 7CQ Is it possible for a car to circle a race track with constant velocity? Can it do so with constant speed? Explain.

Problem 7P CE Predict/Explain You drive your car in a straight line at 15 m/s for 10 kilometers, then at 25 m/s for another 10 kilometers, (a) Is your average speed for the entire trip more than, less than, or equal to 20 m/s? (b) Choose the best explanation from among the following: I. More time is spent at 15 m/s than at 25 m /s. ________________ II. The average of 15 m/s and 25 m/s is 20 m/s. ________________ III. Less time is spent at 15 m/s than at 25 m/s.

Problem 8CQ Friends tell you that on a recent trip their average velocity was +20 m/s. Is it possible that their instantaneous velocity was negative at any time during the trip? Explain.

Problem 8P CE Predict/Explain You drive your car in a Straight line at 15 m/s for 10 minutes, then at 25 m/s for another 10 minutes, (a) Is your average speed for the entire trip more than, less than, or equal to 20m/s?, (b) Choose the best explanation from among the following: I. More time is required fo drive at 15 m/s than at 25 m/s. ________________ II. Less distance is covered at 25 m /s than at 15 m/s. ________________ III. Equal time is spent at 15 m/s and 25 m/s.

Problem 9CQ For what kind of motion are the instantaneous and average velocities equal?

Problem 9P Joseph DeLoach of the United States set an Olympic record in 1988 for the 200-meter dash with a time of 19.75 seconds. What was his average speed? Give your answer in meters per second and miles per hour.

Problem 10CQ If the position of an object is zero, does its speed have to be zero? Explain.

Problem 10P In 1992 Zhuang Yong of China set a women's Olympic record in the 100-meter freestyle swim with a time of 54.64 seconds. What was her average speed in m/s and mi/h?

Problem 11CQ Assume that the brakes in your car create a constant deceleration, regardless of how fast you arc going. If you double your driving speed, how does this affect (a) the time required to come to a stop, and (b) the distance needed to stop?

BIO Kangaroos have been clocked at speeds of 65 km/h. (a) How far can a kangaroo hop in 3.2 minutes at this speed? (b) How long will it take a kangaroo to hop 0.25 km at this speed?

Problem 12CQ The velocity of an object is zero at a given instant of time, (a) Is it possible for the object's acceleration to be zero at this time? Explain, (b) Is it possible for the object's acceleration to be nonzero at this time? Explain.

Problem 12P Rubber Ducks A severe storm on January 10,1992, caused a cargo ship near the Aleutian Islands to spill 29,000 rubber ducks and other bath toys into the ocean. Ten months later hundreds of rubber ducks began to appear along the shoreline near Sitka, Alaska, roughly 1600 miles away. What was the approximate average speed of the ocean current that carried the ducks to shore in (a) m/s and (b) mi/h? (Rubber ducks from the same spill began to appear on the coast of Maine in July 2003.)

Problem 13CQ If the velocity of an object is nonzero, can its acceleration be zero? Give an example if your answer is yes, explain why not if your answer is no.

Problem 13P Radio waves travel at the speed of light, approximately 186,000 miles per second. How long does it take for a radio message to travel from Earth fo the Moon and back? (See the inside back cover for the necessary data.)

Problem 14CQ Is it possible for an object to have zero average velocity over a given interval of time, yet still be accelerating during the interval? Give an example if your answer is yes, explain why not if your answer is no.

Problem 14P It was a dark and stormy night, when suddenly you saw a flash of lightning. Three-and-a-half seconds later you heard the thunder. Given that the speed of sound in air is about 340 m/s, how far away was the lightning bolt?

Problem 15P BIO Nerve Impulses The human nervous system can propagate nerve impulses at about 102 m/s. Estimate the time it takes for a nerve impulse generated when your finger touches a hot object to travel to your brain.

Problem 16CQ A person on a trampoline bounces straight upward with an initial speed of 4.5 m/s. What is the person’s speed when she returns to her initial height?

Problem 16P Estimate how fast your hair grows in miles per hour.

Problem 17CQ After winning a baseball game, one player drops a glove, while another tosses a glove into the air. How do the accelerations of the two gloves compare?

Problem 17P A finch rides on the back of a Galapagos tortoise, which walks at the stately pace of 0.060 m/s. After 1.2 minutes the finch tires of the tortoise's slow pace, and takes flight in the same direction for another 1.2 minutes at 12 m/s. What was the average speed of the finch for this 2.4-minute interval?

Problem 18CQ A volcano shoots a lava bomb straight upward. Does the displacement of the lava bomb depend on (a) your choice of origin for your coordinate system, or (b) your choice of a positive direction? Explain in each case.

Problem 18P You jog at 9.5 km/h for 8.0 km, then you jump into a car and drive an additional 16 km. With what average speed must you drive your car if your average speed for the entire 24 km is to be 22 km/h?

A dog runs back and forth between its two owners, who are walking toward one another (Figure 2–24). The dog starts running when the owners are 10.0 m apart. If the dog runs with a speed of 3.0 m/s, and the owners each walk with a speed of 1.3 m/s, how far has the dog traveled when the owners meet?

Problem 20P IP You drive in a straight line at 20.0 m/s for 10.0 minutes, then at 30.0 m/s for another 10.0 minutes, (a) Is your average speed 25.0 m/s, more than 25.0 m/s, or less than 25.0 m/s? Explain, (b) Verify your answer to part (a) by calculating the average speed.

Problem 21P In heavy rush-hour traffic you drive in a straight line at 12m/s for 1.5 minutes, then you have to stop for 3.5 minutes, and finally you drive at 15 m/s for another 2.5 minutes, (a) Plot a position-versus-time graph for this motion. Your plot should extend from t = 0 to t = 7.5 minutes, (b) Use your plot from part (a) to calculate the average velocity between t = 0 and t = 7.5 minutes.

Problem 22P IP You drive in a straight line at 20.0 m/s for 10.0 miles, then at 30.0 m/s for another 10.0 miles, (a) Is your average speed 25.0 m/s, more than 25.0 m/s, or less than 25.0 m/s? Explain, (b) Verify your answer to part (a) by calculating the average speed.

Problem 24P The position of a particle as a function of time is given by x = (-5 m/s)t + (3 m/s2)t2. (a) Plot x d versus t for t = 0 to t = 2s. (b) Find the average velocity of the particle from t = 0 to t = 1 s. (c) Find the average speed from t = 0 to t = 1 s.

IP An expectant father paces back and forth, producing the position- versus-time graph shown in Figure 2–25. Without performing a calculation, indicate whether the father’s velocity is positive, negative, or zero on each of the following segments of the graph: (a) A, (b) B, (c) C, and (d) D. Calculate the numerical value of the father’s velocity for the segments (e) A, (f) B, (g) C, and (h) D, and show that your results verify your answers to parts (a)–(d).

Problem 25P The position of a particle as a function of time is given by x = (6 m/s)t + (-2 m/s2)t2. (a) Plot x versus t for t = 0 to t = 2 s. (b) Find the average velocity of the particle from t = 0 to t = 1 s. (c) Find the average speed from t = 0 to t = 1 s.

Problem 27P On your wedding day you leave for the church 30.0 minutes before the ceremony is to begin, which should be plenty of time since the church is only 10.0 miles away. On the way, however, you have to make an unanticipated stop for construction work on the road. As a result, your average speed for the first 15 minutes is only 5.0 mi/h. What average speed do you need for the rest of the trip to get you to the church on time?

CE The position-versus-time plot of a boat positioning itself next to a dock is shown in Figure 2–27. Rank the six points indicated in the plot in order of increasing value of the velocity ????, starting with the most negative. Indicate a tie with an equal sign.

IP A tennis player moves back and forth along the baseline while waiting for her opponent to serve, producing the position-versus- time graph shown in Figure 2–26. (a)Without performing a calculation, indicate on which of the segments of the graph, A, B, or C, the player has the greatest speed. Calculate the player’s speed for (b) segment A, (c) segment B, and (d) segment C, and show that your results verify your answers to part (a).

Problem 30P The position of a particle as a function of time is given by x = (-2.00 m/s)t + (3.00 m/s3)t3. (a) Plot x versus t for time from t = 0 to t = 1.00 s. (b) Find the average velocity of the particle from t = 0.150 s to t = 0.250 s. (c) Find the average velocity from t = 0.190 s to t = 0.210 s. (d) Do you expect the instantaneous velocity at t = 0.200 s to be closer to -1.62 m/s, or -1.66 m/s? Explain.

Problem 29P The position of a particle as a function of time is given by x = (2.0 m/s)t + (-3.0 m/s3)t3. (a) Plot x versus t for time from t = 0 to t = 1,0 s. (b) Find the average velocity of the particle from t = 0.35 s to t = 0.45 s. (c) Find the average velocity from t = 0.39 s to t = 0.41 s. (d) Do you expect the instantaneous velocity at t = 0.40 s to be closer to 0.54 m/s, 0.56 m/s, or 0.58 m/s? Explain.

Problem 31P CE Predict/Explain Two bows shoot identical arrows with the same launch speed. To accomplish this, the string in bow 1 must be pulled back farther when, shooting its arrow than the string in bow 2. (a) Is the acceleration of the arrow shot by bow 1 greater than, less than, or equal to the acceleration of the arrow shot by bow 2? (b) Choose the best explanation from among the following: I. The arrow in bow 2 accelerates for a greater time. II. Both arrows start from rest. III. The arrow in bow 1 accelerates for a greater time.

Problem 34P A jet makes a landing traveling due east with a speed of 115 m/s. If the jet comes to rest in 13.0 s, what are the magnitude and direction of its average acceleration?

Problem 32P A 747 airliner reaches its takeoff speed of 173 mi/h in 35.2 s. What is the magnitude of its average acceleration?

Problem 33P At the starling gun, a runner accelerates at 1.9 m/s2 for 5.2 s. The runner's acceleration is zero for the rest of the race. What is the speed of the runner (a) at t = 2.0 s, and (b) at the end of the race?

Problem 35P A car is traveling due north at 18.1 m/s. Find the velocity of the car after 7.50 s if its acceleration is (a) 1.30 m/s2 due north, or (b) 1.15 m/s2 due south.

Problem 38P Running with an initial velocity of +11 m/s, a horse has an average acceleration of -1.81 m/s2. How long does it take for the horse to decrease its velocity to +6.5 m/s?

Problem 40P IP In the previous problem, (a) does the distance needed to stop increase by a factor of two or a factor of four? Explain. Verify your answer to, part (a) by calculating the stopping distances for initial speeds of (b) 16 m/s and (c) 32 m/s.

Problem 39P IP Assume that the brakes in your car create a constant deceleration of 4.2 m/s2 regardless of how fast you are driving. If you double your driving speed from 16 m/s to 32 m/s, (a) does the time required to come to a stop increase by a factor of two or a factor of four? Explain. Verify your answer to part (a) by calculating the stopping limes for initial speeds of (b) 16 m/s and (c) 32 m/s.

A person on horseback moves according to the velocity-versus- time graph shown in Figure 2–29. Find the displacement of the person for each of the following segments of the motion: (a) A, (b) B, and (c) C.

As a train accelerates away from a station, it reaches a speed of 4.7 m/s in 5.0 s. If the train's acceleration remains constant, what is its speed after an additional 6.0 s has elapsed?

Problem 43P Landing with a speed of 81.9 m/s, and traveling due south, a jet comes to rest in 949 m. Assuming the jet slows with constant acceleration, find the magnitude and direction of its acceleration.

Problem 42P A particle has an acceleration of +6.24 m/s2 for 0.300 s. At the end of this time the particle's velocity is +9.31 m/s. What was the particle's initial velocity?

Problem 44P When you see a traffic light turn red, you apply the brakes until you come to a stop. If your initial speed was 12 m/s, and you were heading due west, what was your average velocity during braking? Assume constant deceleration.

Problem 45P A ball is released at the point x = 2 m on an inclined plane with a nonzero initial velocity. After being released, the ball moves with constant acceleration. The acceleration and initial velocity of the ball are described by one of the following four cases: case 1, a > 0, v0 > 0; case 2, a > 0, v0 < 0; case 3, a < 0, v0 > 0; case 4, a < 0, v0 < 0. (a) In which of these cases will the ball definitely pass x = 0 at some later time? (b) In which of these cases is more information needed to determine whether the ball will cross x = 0? (c) In which of these cases will the ball come to rest momentarily at some time during its motion?

Problem 47P Starting from rest, a boat increases its speed to 4.12 m/s with constant acceleration, (a) What is the boat's average speed? (b) If it takes the boat 4.77 s to reach this speed, how far has it traveled?

Problem 48P IP BIO A cheetah can accelerate from rest to 25.0 m/s in 6.22 s. Assuming constant acceleration, (a) how far has the cheetah run in this time? (b) After sprinting for just 3.13 s, is the cheetah's speed 12.5 m/s, more than 12.5 m/s, or less than 12.5 m/s? Explain, (c) What is the cheetah's average speed for the first 3.11 s of its sprint? For the second 3.11 s of its sprint? (d) Calculate the distance covered by the cheetah in the first 3.11 s and the second 3.11 s.

Problem 49P A child slides down a hill on a toboggan with an acceleration of 1.8 m/s2. If she starts at rest, how far has she traveled in (a) 1.0 s, (b) 2.0 s, and (c) 3.0 s?

Problem 46P Suppose the car in Problem 44 comes to rest in 35 m. How much time does this take?

Problem 51P Air Bags Air bags are designed to deploy in 10 ms. Estimate the acceleration of the front surface of the bag as it expands. Express your answer in terms of the acceleration of gravity g.

A motorcycle moves according to the velocity-versus-time graph shown in Figure 2–28. Find the average acceleration of the motorcycle during each of the following segments of the motion: (a) A, (b) B, and (c) C.

Problem 53P BIO Bacterial Motion Approximately 0,1% of the bacteria in an adult human's intestines are Escherichia coli. These bacteria have been observed to move with speeds up to 15 µm/s and maximum accelerations of 166 µm/s2. Suppose an E. coli bacterium in your intestines starts at rest and accelerates at 156 µm/s2. How much (a) time and (b) distance are required for the bacterium to reach a speed of 12 µm/s?

Problem 52P Jules Verne In his novel From the Earth to the Moon (1866), Jules Verne describes a spaceship that is blasted out of a cannon, called the Columbiad, with a speed of 12,000 yards/s. The Columbiad is 900 ft long, but part of it is packed with powder, so the spaceship accelerates over a distance of only 700 ft. Estimate the acceleration experienced by the occupants of the spaceship during launch. Give your answer in m/s2. (Verne realized that the "travelers would … encounter a violent recoil," but he probably didn't know that people generally lose consciousness if they experience accelerations greater than about 7g ~ 70 m/s2.)

The Detonator On a ride called the Detonator at Worlds of Fun in Kansas City, passengers accelerate straight downward from rest to 45 mi/h in 2.2 seconds. What is the average acceleration of the passengers on this ride?

Problem 54P Two cars drive on a straight highway. At time t = 0, car 1 passes mile marker 0 traveling due east with a speed of 20.0 m/s. At the same time, car 2 is 1.0 km east of mile marker 0 traveling at 30.0 m/s due west. Car 1 is speeding up with an acceleration of magnitude 2.5 m/s2, and car 2 is slowing down with an acceleration of magnitude 3.2 m/s2, (a) Write x-versus-t equations of motion for both cars, taking east as the positive direction, (b) At what time do the cars pass next to one another?

IP You are driving through town at \(12.0 \mathrm{~m} / \mathrm{s}\) when suddenly a ball rolls out in front of you. You apply the brakes and begin decelerating at \(3.5 \mathrm{~m} / \mathrm{s}^{2}\). (a) How far do you travel before stopping? (b) When you have traveled only half the distance in part (a), is your speed \(6.0 \mathrm{~m} / \mathrm{s}\), greater than \(6.0 \mathrm{~m} / \mathrm{s}\), or less than \(6.0 \mathrm{~m} / \mathrm{s}\)? Support your answer with a calculation.

A rocket blasts off and moves straight upward from the launch pad with constant acceleration. After 3.0 s the rocket is at a height of 77 m. (a) What are the magnitude and direction of the rocket's acceleration? (b) What is its speed at this time?

Problem 55P A Meteorite Strikes On October 9, 1992, a 27-pound meteorite struck a car in Peekskill, NY, leaving a dent 22 cm deep in the trunk. If the meteorite struck the car with a speed of 130 m/s, what was the magnitude of its deceleration, assuming it to be constant?

Problem 58P IP You are driving through town at 16 m/s when suddenly a car backs out of a driveway in front of you. You apply the brakes and begin decelerating at 3.2 m/s2, (a) How much time does it take to stop? (b) After braking half the time found in part (a), is your speed 8.0 m/s, greater than 8.0 m/s, or less than 8.0 m/s? Support your answer with a calculation, (c) If the car backing out was initially 55 m in front of you, what is the maximum reaction time you can have before hitting the brakes and still avoid hitting the car?

IP BIO A Tongue’s Acceleration When a chameleon captures an insect, its tongue can extend 16 cm in 0.10 s. (a) Find the magnitude of the tongue’s acceleration, assuming it to be constant. (b) In the first 0.050 s, does the tongue extend 8.0 cm, more than 8.0 cm, or less than 8.0 cm? Support your conclusion with a calculation.

Problem 60P IP Coasting due west on your bicycle at 8.4 m/s, you encounter a sandy patch of road 7.2 m across. When you leave the sandy patch your speed has been reduced by 2.0 m/s to 6.4 m/s. (a) Assuming the sand causes a constant acceleration, what was the bicycle's acceleration in the sandy patch? Give both magnitude and direction, (b) How long did it take to cross the sandy patch? (c) Suppose you enter the sandy patch with a speed of only 5.4 m /s. Is your final speed in this case 3.4 m/s, more than 3.4 m/s, or less than 3.4 m/s? Explain.

Problem 62P IP A boat is cruising in a straight line at a constant speed of 2.6 m/s when it is shifted into neutral. After coasting 12 m the engine is engaged again, and the boat resumes cruising at the reduced constant speed of 1.6 m/s. Assuming constant acceleration while coasting, (a) how long did it take for the boat to coast the 12 m? (b) What was the boat's acceleration while it was coasting? (c) When the boat had coasted for 6.0 m, was its speed 2.1 m/s, more than 2.1 m/s, or less than 2.1 m/s? Explain.

Problem 61P BIO Surviving a Large Deceleration On July 13, 1977, while on a test drive at Britain's Silverstone racetrack, the throttle on David Purley's car stuck wide open. The resulting crash subjected Purley to the greatest "g-force" ever survived by a human—he decelerated torn 173 km/h to zero in a distance of only about 0.66 m. Calculate the magnitude of the acceleration experienced by Purley (assuming it to be constant), and express your answer in units of the acceleration of gravity, g = 9.81 m/s2.

Problem 63P A model rocket rises with constant acceleration to a height of 3.2 m, at which point its speed is 26.0 m/s. (a) How much time does it take for the rocket to reach this height? (b) What was the magnitude of the rocket's acceleration? (c) Find the height and speed of the rocket 0.10 s after launch.

Problem 68P In a physics lab, students measure the time it takes a small cart to slide a distance of 1.00 m on a smooth track inclined at an angle ? above the horizontal. Their results are given in the following table. ? 10.0° 20.0° 30.0° time, s 1.08 0.770 0.640 (a) Find the magnitude of the cart's acceleration for each angle. ________________ (b) Show that your results for part (a) are in close agreement with the formula, a = g sin ?. (We will derive this formula in Chapter 5.)

A car in stop-and-go traffic starts at rest, moves forward 13 m in 8.0 s, then comes to rest again. The velocity-versus-time plot for this car is given in Figure 2–30. What distance does the car cover in (a) the first 4.0 seconds of its motion and (b) the last 2.0 seconds of its motion? (c) What is the constant speed ???? that characterizes the middle portion of its motion?

Problem 69P CE At the edge of a roof you throw ball an initial speed u0;a moment later you throw ball 2 downward with the same initial speed. The balls land at the same time. Which of the following statements is true for the instant just before the balls hit the ground? A. The speed of ball 1 is greater than the speed of ball 2; B. The speed of ball 1 is equal to the speed of ball 2; C. The speed of ball 1 is less than the speed of ball 2. 70. Legend has it that Isaac Newton was hit on the head by a falling apple, thus triggering Ms thoughts on gravity. Assuming the story to be true, estimate the speed of the apple when it struck Newton.

A car and a truck are heading directly toward one another on a straight and narrow street, but they avoid a head-on collision by simultaneously applying their brakes at \(t=0\). The resulting velocity-versus-time graphs are shown in Figure . What is the separation between the car and the truck when they have come to rest, given that at \(t=0\) the car is at \(x=15\ \mathrm m\) and the truck is at \(x=-35\ \mathrm m\)? (Note that this information determines which line in the graph corresponds to which vehicle.) ________________ Equation Transcription: Text Transcription: t=0 t=0 x=15 m x=-35 m

A bicyclist is finishing his repair of a flat tire when a friend rides by with a constant speed of 3.5 m/s. Two seconds later the bicyclist hops on his bike and accelerates at \(2.4\mathrm{\ m}/\mathrm{s}^2\) until he catches his friend. (a) How much time does it take until he catches his friend? (b) How far has he traveled in this time? (c) What is his speed when he catches up?

Problem 64P The infamous chicken is dashing toward home plate with a speed of 5.8 m/s when he decides to hit the dirt. The chicken slides for 1.1 s, just reaching the plate as he stops (safe, of course), (a) What are the magnitude and direction of the chicken's acceleration? (b) How far did the chicken slide?

Problem 72P Referring to the cartoon in Problem 71, how long would it take For the car to go from 0 to 30 mi/h?

Problem 70P Legend has it that Isaac Newton was hit on the head by a falling apple, thus triggering his thoughts on gravity. Assuming the story to be true, estimate the speed of the apple when it struck Newton.

Problem 71P The cartoon shows a car in free fall. Is the statement made in the cartoon accurate? Justify your answer.

Problem 76P An Extraterrestrial Volcano The first active volcano observed outside the Earth was discovered in 1979 on lo, one of the moons of Jupiter. The volcano was observed to be ejecting material to a height of about 2.00 x 105 m,Given that the acceleration of gravity on lo is 1.80 m/s2, find the initial velocity of the ejected material.

BIO Measure Your Reaction Time Here's something you can try at home—an experiment to measure your reaction time. Have a friend hold a ruler by one end, letting the other end hang down vertically. At the lower end, hold your thumb and index finger on either side of the ruler, ready to grip it. Have your friend release the ruler without warning. Catch it as quickly as you can. If you catch the ruler 5.2 cm from the lower end, what is your reaction time?

CE Predict/Explain A carpenter on the roof of a building accidentally drops her hammer. As the hammer falls it passes two windows of equal height, as shown in Figure 2–32. (a) Is the increase in speed of the hammer as it drops past window 1 greater than, less than, or equal to the increase in speed as it drops past window 2? (b) Choose the best explanation from among the following: I. The greater speed at window 2 results in a greater increase in speed. II. Constant acceleration means the hammer speeds up the same amount for each window. III. The hammer spends more time dropping past window 1.

CE Predict/Explain Figure 2–33 shows a ????-versus-???? plot for the hammer dropped by the carpenter in Problem 78. Notice that the times when the hammer passes the two windows are indicated by shaded areas. (a) Is the area of the shaded region corresponding to window 1 greater than, less than, or equal to the area of the shaded region corresponding to window 2? (b) Choose the best explanation from among the following: I. The shaded area for window 2 is higher than the shaded area for window 1. II. The windows are equally tall. III. The shaded area for window 1 is wider than the shaded area for window 2.

Problem 74P BIO Gulls are often observed dropping clams and other shellfish from a height to the rocks below, as a means of opening the shells. If a seagull drops a shell from rest at a height of 14 m, how fast is the shell moving when it hits the rocks? 75. A volcano launches a lava bomb straight upward with an initial speed of 28 m/s. Taking upward to be the positive direction, find the speed and direction of motion of the lava bomb (a) 2.0 seconds and (b) 3.0 seconds after it is launched.

CE A ball is thrown straight upward with an initial speed . When it reaches the top of its flight at height , a second ball is thrown straight upward with the same initial speed. Do the balls cross paths at height \(\frac{1}{2}h\), above \(\frac{1}{2}h\), or below \(\frac{1}{2}h\)? ________________ Equation Transcription: Text Transcription: frac{1}{2}h frac{1}{2}h frac{1}{2}h

Problem 81P Bill steps off a 3.0-m-high diving board and drops to the water below. At the same time, Ted jumps upward with a speed of 4.2 m/s from a 1.0-m-high diving board. Choosing the origin to be at the water's surface, and upward to be the positive x direction, write x-versus-t equations of motion for both Bill and Ted.

Problem 84P Highest Water Fountain The world's highest fountain of water is located, appropriately enough, in Fountain Hills, Arizona. The fountain rises to a height of 560 ft (5 feet higher than the Washington Monument), (a) What is the initial speed of the water? (b) How long does it take for water to reach the top of the fountain?

Problem 82P Repeat the previous problem, this time with the origin. 3.0 m above the water, and with downward as the positive x direction.

Problem 73P Jordan's Jump Michael Jordan's vertical leap is reported to be 48 inches. What is his takeoff speed? Give your answer m meters per second.

Problem 83P On a hot summer day in the state of Washington while kayaking, I saw several swimmers jump from a railroad bridge into the Snohomish River below, The swimmers stepped off the bridge, and I estimated that they hit the water 1.5 s later, (a) How high was the bridge? (b) How fast were the swimmers moving when they hit the water? (c) What would the swimmers' drop time be if the bridge were twice as high?

You shoot an arrow into the air. Two seconds later (2.00 s) the arrow has gone straight upward to a height of 30.0 m above its launch point, (a) What was the arrow's initial speed? (b) How long did it take for the arrow to first reach a height of 15.0 m above its launch point?

Standing at the edge of a cliff 32.5 m high, you drop a ball. Later, you throw a second ball downward with an initial speed of 11.0 m/s. (a) Which ball has the greater increase in speed when it reaches the base of the cliff, or do both balls speed up by the same amount? (b) Verify your answer to part (a) with a calculation.

Problem 85P Wrongly called for a foul, an angry basketball player throws the ball straight down to the floor. If the ball bounces straight up and returns to the floor 2.8 s after first striking it, what was the ball's greatest height above the floor?

Problem 89P While riding on an elevator descending with a constant speed of 3.0 m/s, you accidentally drop a book from under your arm. (a) How long does it take for the book to reach the elevator floor, 1.2 m below your arm? (b) What is the book's speed relative to you when it hits the elevator floor?

Problem 92P A model rocket blasts off and moves upward with an acceleration of 12 m/s2 until it reaches a height of 26 m, at which point its engine shuts off and it continues its flight in free fall. (a) What is the maximum height attained by the rocket? ________________ (b) Whatis the speed of the rocket just before it hits the ground? ________________ (c) What is the total duration of the rocket's flight?

A hot-air balloon is descending at a rate of 2.0 m/s when a passenger drops a camera. If the camera is 45 m above the ground when it is dropped, (a) how long does it take for the camera to reach the ground, and (b) what is its velocity just before it lands? Let upward be the positive direction for this problem.

Problem 91P IP Standing side by side, you and a friend step off a bridge at different times and fall for 1.6 s to the water below. Your friend goes first, and you follow after she has dropped a distance of 2.0 m. (a) When your friend, hits the water, is the separation between the two of you 2.0 m, less than 2.0 m, or more than 2.0 m? (b) Verify your answer to part (a) with a calculation.

Hitting the “High Striker” A young woman at a carnival steps up to the “high striker,” a popular test of strength where the contestant hits one end of a lever with a mallet, propelling a small metal plug upward toward a bell. She gives the mallet a mighty swing and sends the plug to the top of the striker, where it rings the bell. Figure 2–34 shows the corresponding position-versus-time plot for the plug. Using the in- formation given in the plot, answer the following questions: (a) What is the average speed of the plug during its upward journey? (b) By how much does the speed of the plug decrease during its upward journey? (c) What is the initial speed of the plug? (Assume the plug to be in free fall during its upward motion, with no effects of air resistance or friction.)

In a well-known Jules Verne novel, Phileas Fogg travels around the world in 80 days. What was Mr. Fogg's approximate average speed during his adventure?

Problem 94P While sitting on a tree branch 10.0 m above the ground, you drop a chestnut. When the chestnut has fallen 2.5 m, you throw a second chestnut straight down. What initial speed must you give the second chestnut if they are both to reach the ground at the same time?

Problem 96GP An astronaut on the Moon drops a rock straight downward from a height of 1.25 m. If the acceleration of gravity on the Moon is 1.62 m/s2, what is the speed of the rock just before it lands?

Problem 98GP A Supersonic Waterfall Geologists have learned of periods in the past when the Strait of Gibraltar closed off, and the Mediterranean Sea dried out and become a desert. Later, when the strait reopened, a massive saltwater waterfall was created. According to geologists, the water in this waterfall was supersonic; that is, it fell with speeds in excess of the speed of sound. Ignoring air resistance, what is the minimum height necessary to create a supersonic waterfall? (The speed of sound may be taken to be 340 m/s.)

CE Suppose the two balls described in Problem 99 are released at the same time, with ball Adropped from rest and ball B thrown downward with an initial speed \(v_0\). Identify which of the five plots shown in Figure 2–35 corresponds to (a) ball A and (b) ball B. ________________ Equation Transcription: Text Transcription: v_0 v_0

Problem 97GP You jump from the top of a boulder to the ground 1.5 m below. Estimate your deceleration on landing.

Problem 104GP At the 18th green of the U.S. Open you need to make a 20.5-ft putt to win tie tournament. When you hit the ball, giving it an initial speed of 1.57 m/s, it stops 6.00 ft short of the hole. (a) Assuming the deceleration caused by the grass is constant, what should the initial speed have been to just make the putt? (b) What initial speed do you need to make the remaining 6.00-ft putt?

CE At the edge of a roof you drop ball A from rest, and then throw ball B downward with an initial velocity of \(v_0\). Is the increase in speed just before the balls land more for ball A, more for ball B, or the same for each ball? ________________ Equation Transcription: Text Transcription: v_0

Referring to Conceptual Checkpoint , find the separation between the rocks at (a) \(t=1.0\ \mathrm{s}\), (b) \(t=2.0\ \mathrm{s}\), and (c) \(t=3.0\ \mathrm{s}\), where time is measured from the instant the second rock is dropped. (d) Verify that the separation increases linearly with time. ________________ Equation Transcription: Text Transcription: t=1.0 s t=2.0 s t=3.0 s

Problem 105GP A popular entertainment at some carnivals is the blanket toss (see photo, p. 39). (a) If a person is thrown to a maximum height of 28.0 ft above the blanket, how long does she spend in the air? (b) Is the amount of time the person is above a height of 14.0 ft more than, less than, or equal to the amount of time the person is below a height of 14.0 ft? Explain. (c) Verify your answer to part (b) with a calculation.

Problem 102GP Drop Tower NASA operates a 2.2-second drop tower at the Glenn Research Center in Geveland, Ohio. At this facility, experimental packages are dropped from the top of the tower, on the 8th floor of the building. During their 2.2 seconds of free fall, experiments experience a microgravity environment similar to that of a spacecraft in orbit. (a) What is the drop distance of a 2.2-s tower? (b) How fast are the experiments traveling when they hit the air bags at the bottom of the tower? (c) If the experimental package comes to rest over a distance of 0.75 m upon hitting the air bags, what is the average stopping acceleration?

Problem 103GP A youngster bounces straight up and down on a trampoline. Suppose she doubles her initial speed from 2.0 m/s to 4.0 m/s. (a) By what factor does her time in the air increase? (b) By what factor does her maximum height increase? (c) Verify your answers to parts (a) and (b) with an explicit calculation.

A glaucous-winged gull, ascending straight upward at 5.20 m/s, Tops a shell when it is 12.5 m above the ground. (a) What are the magnitude and direction of the shell’s acceleration just after it is released? (b) Find the maximum height above the ground reached by the shell. (c) How long does it take for the shell to reach the ground? (d) What is the speed of the shell at this time?

Astronauts on a distant planet throw a rock straight upward and record its motion with a video camera. After digitizing their video, they are able to produce the graph of height, ????, versus time, ????, shown in Figure 2–36. (a) What is the acceleration of gravity on this planet? (b) What was the initial speed of the rock?

Problem 108GP A doctor, preparing to give a patient an injection, squirts a small amount of liquid straight upward from a syringe. If the liquid emerges with a speed of 1.5 m/s, (a) how long does it take for it to return to the level of the syringe? (b) What is the maximum height of the liquid above the syringe?

Problem 117GP To find the height of an overhead power line, you throw a ball straight upward. The ball passes the line on the way up after 0.75 s, and passes it again on the way down 1.5 s after it was tossed. What are the height of the power line and the initial speed of the ball?

Problem 113GP Weights are tied to each end of a 20.0-cm string. You hold one weight in your hand and let the other hang vertically a height h above the floor. When you release the weight in your hand, the two weights strike the ground one after the other with audible thuds. Find the value of h for which the time between release and the first thud is equal to the time between the first thud and the second thud.

Suppose the first rock in Conceptual Checkpoint drops through a height before the second rock is released from rest. Show that the separation between the rocks, , is given by the following expression: \(S=h+(sqrt{2gh})t\) In this result, the time is measured from the time the second rock is dropped. ________________ Equation Transcription: Text Transcription: S=h+(sqrt{2gh})t

A ball, dropped from rest, covers three-quarters of the distance to the ground in the last second of its fall. (a) From what height was the ball dropped? (b) What was the total time of fall?

Problem 115GP A stalactite on the roof of a cave drips water at a steady rate to a pool 4.0 m below. As one drop of water hits the pool, a second drop is in the air, and a third is just detaching from the stalactite. (a) What are the position and velocity of the second drop when the first drop hits the pool? (b) How many drops per minute fall into the pool?

Problem 119GP An arrow is fired with a speed of 20.0 m/s at a block of Styrofoam resting on a smooth surface. The arrow penetrates a certain distance into the block before coining to rest relative to it. During this process the arrow’s deceleration has a magnitude of 1550 m/s2 and the block’s acceleration has a magnitude of 450 m/s2. (a) How long does it take for the arrow to stop moving with rcspeet to the block? (b) What is the common speed of the arrow and block when this happens? (c) How far into the block does the arrow penetrate?

Referring to Example 2–9 Suppose the speeder (red car) is traveling with a constant speed of 25 m/s, and that the maximum acceleration of the police car (blue car) is \(3.8\ \mathrm{ m/s^2}\). If the police car is to start from rest and catch the speeder in 15 s or less, what is the maximum head-start distance the speeder can have? Measure time from the moment the police car starts. ________________ Equation Transcription: Text Transcription: 3.8 m/s^2

Problem 120GP Sitting in a second-story apartment, a physicist notices a ball moving straight upward just outside her window. The ball is visible for 0.25 s as it moves a distance of 1.05 m from the bottom to the top of the window. (a) How long does it take before the ball reappears? (b) What is the greatest height of the ball above the top of the window?

Referring to Example 2–9 The speeder passes the position of the police car with a constant speed of 15 m/s. The police car immediately starts from rest and pursues the speeder with constant acceleration. What acceleration must the police car have if it is to catch the speeder in 7.0 s? Measure time from the moment the police car starts.

Suppose, instead of shutting off the engine, the astronauts had increased its thrust, giving the lander a small, but constant, upward acceleration. Which speed-versus-time plot in Figure 2–37 (b) would describe this situation? A B C D

Problem 116GP You drop a ski glove from a height h onto fresh snow, and it sinks to a depth d before coming to rest. (a) In terms of g and h, what is the speed of the glove when it reaches the snow? (b) What are the magnitude and direction of the glove’s acceleration as it moves through the snow, assuming it to be constant? Give your answer in terms of g, h, and d.

Which of the speed-versus-time plots in Figure 2–37 (b) correctly represents the speed of the Apollo 15 lander? A B C D

IP Referring to Example 2–12 (a) In Example 2–12, the bag of sand is released at 20.0 m and reaches a maximum height of 22 m. If the bag had been released at 30.0 m instead, with everything else remaining the same, would its maximum height be 32 m, greater than 32 m, or less than 32 m? (b) Find the speed of the bag just before it lands when it is released from 30.0 m.

A volcano launches a lava bomb straight upward with an initial speed of 28 m/s. Taking upward to be the positive direction, find the speed and direction of motion of the lava bomb (a) 2.0 seconds and (b) 3.0 seconds after it is launched.

Problem 86P To celebrate a victory, a pitcher throws her glove straight upward with an initial speed of 6.0 m/s. (a) How long does it take for the glove to return to the pitcher? (b) How long does it take for the glove to reach its maximum height?

Problem 110GP In the previous problem, what is the minimum initial speed of the camera if it is to just reach the passenger? (Hint: When the camera is thrown with its minimum speed, its speed on reaching the passenger is the same as the speed of the passenger.)

Problem 109GP A hot-air balloon has just lifted off and is rising at the constant rate of 2.0 m/s. Suddenly one of the passengers realizes she has left her camera on the ground. A friend picks it up and tosses it straight upward with an initial speed of 13 m/s. If the passenger is 2.5 m above her friend when the camera is tossed, how high is she when the camera reaches her?

Problem 111GP Old Faithful Watching Old Faithful erupt, you notice that it takes a time t for water to emerge from the base of the geyser and reach its maximum height. (a) What is the height of the geyser, and (b) what is the initial speed of the water? Evaluate your expressions for (c) the height and (d) the initial speed for a measured time of 1.65 s.

Problem 112GP A ball is thrown upward with an initial speed ?0. When it reaches the top of its flight, at a height h, a second ball is thrown upward with the same initial velocity. (a) Sketch an x-versus-t plot for each ball. (b) From your graph, decide whether the balls cross paths at h/2, above h/2, or below h/2. (c) Find the height where the paths cross.

The Quadratic Formula from Kinematics In this problem we show how the kinematic equations of motion can be used to derive the quadratic formula. First, consider an object with an initial position \(x_0\), an initial velocity \(v_0\), and an acceleration . To find the time when this object reaches the position \(x=0\) we can use the quadratic formula, or apply the following two-step procedure: (a) Use Equation to show that the velocity of the object when it reaches \(x=0\) is given by \(v=\pm\sqrt{v_0^2-2ax_0}\). (b) Use Equation to show that the time corresponding to the velocity found in part (a) is \(t=\frac{-v_0\pm\sqrt{v_0^2-2ax_0}}{a}\). (c) To complete our derivation, show that the result of part (b) is the same as applying the quadratic formula to \(x=x_0+v_0t+\frac{1}{2}at^2=0\). ________________ Equation Transcription: Text Transcription: x_0 v_0 x=0 x=0 v={+/-}sqrt{v_0^2-2ax_0} t=frac{-v_0{+/-}sqrt{v_0^2-2ax0}}{a} x=x_0+v_0t+frac{1}{2}at^{2}=0

How long did it take for the lander to drop the final 4.30 ft to the Moon’s surface? A. 1.18 s B. 1.37 s C. 1.78 s D. 2.36 s

Problem 15CQ A batter hits a pop fly straight up. (a) Is the acceleration of the ball on the way up different from its acceleration on the way down? (b) Is the acceleration of the ball at the top of its flight different from its acceleration just before it lands?

What was the impact speed of the lander when it touched down? Give your answer in feet per second (ft/s), the same units used by the astronauts. A. \(2.41 \mathrm{ft} / \mathrm{s}\) B. \(6.78 \mathrm{ft} / \mathrm{s}\) C. \(9.95 \mathrm{ft} / \mathrm{s}\) D. \(10.6 \mathrm{ft} / \mathrm{s}\) Equation Transcription: Text Transcription: 2.41 ft/s 6.78 ft/s 9.95 ft/s 10.6 ft/s