Master the Concepts

Essential relationships for constant acceleration problems: If

*a*_{x}is constant during the entire time interval Δ*t*from*t*_{i}until a later time*t*_{f}=*t*_{i}+ Δ*t,*(4-1)(4-3)(4-4)(4-5)These same relationships hold for the

*y*-components of the position, velocity, and acceleration if*a*_{y}is constant.The only force acting on an object in free fall is gravity. On Earth, free fall is an idealization since there is always

*some*air resistance. An object in free fall has an acceleration equal to the local value of the gravitational fieldFor a projectile or any object moving with constant acceleration in the ±

*y*-direction, the motion in the*x*- and*y*-directions can be treated separately. Since*a*_{x}= 0,*v*_{x}is constant. Thus, the motion is a superposition of constant velocity motion in the*x*-direction and constant acceleration motion in the*y*-direction.The

*x*- and*y*-axes are chosen to make the problem easiest to solve. Any choice is valid as long as the two are perpendicular. In an equilibrium problem, choose*x*- and*y*-axes so that the fewest number of force vectors have to be resolved into both*x*- and*y*-components. In a nonequilibrium problem, if the direction of the acceleration is known, choose*x-*and*y*-axes so that the acceleration vector is parallel to one of the axes.Problems involving Newton's second law—whether equilibrium or nonequilibrium—can be solved by treating the

*x*- and*y*-components of the forces and the acceleration separately.An object that is accelerating has an apparent weight that differs from its true weight. The apparent weight is equal to the normal force exerted by a supporting surface with the same acceleration. A helpful trick is to think of the apparent weight as the reading of a bathroom scale that supports the object.

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Conceptual Questions

Why is the muzzle of a rifle not aimed directly at the center of the target?

Can the velocity of an object be zero and the acceleration be nonzero at the same time? Explain.

Does the monkey, coconut, and hunter demonstration still work if the arrow is pointed

*downward*at the monkey and coconut? Explain.Can a body in free fall be in equilibrium? Explain.

If a feather and a lead brick are dropped simultaneously from the top of a ladder, the lead brick hits the ground first. What would happen if the experiment is repeated on the surface of the Moon?

Why does a 1-kg sandbag fall with the same acceleration as a 5-kg sandbag? Explain in terms of Newton's second law and his law of gravitation.

An object is placed on a scale. Under what conditions does the scale read something other than the object's weight, even though the scale is functioning properly and is calibrated correctly? Explain.

What does it mean when we refer to a cord as an “ideal cord” and a pulley as an “ideal pulley”?

What force(s) act on a parachutist descending to Earth with a constant velocity? What is the acceleration of the parachutist?

Is it possible for two identical projectiles with identical initial speeds, but with two different angles of elevation, to land in the same spot? Explain. Ignore air resistance and sketch the trajectories.

A baseball is tossed straight up. Taking into consideration the force of air resistance, is the magnitude of the baseball's acceleration zero, less than

*g,*equal to*g,*or greater than*g*on the way up? At the top of the flight? On the way down? Explain. [*Hint:*The force of air resistance is directed opposite to the velocity. Assume in this case that its magnitude is less than the weight.]What is the acceleration of an object thrown straight up into the air at the highest point of its motion? Does the answer depend on whether air resistance is negligible or not? Explain.

If the trajectory is parabolic in one reference frame, is it always, never, or sometimes parabolic in another reference frame that moves at constant velocity with respect to the first reference frame? If the trajectory can be other than parabolic, what else can it be?

If air resistance is ignored, what force(s) act on an object in free fall?

Why might an elevator cable break during acceleration when lifting a lighter load than it normally supports at rest or at constant velocity?

You are standing on a balcony overlooking the beach. You throw a ball straight up into the air with speed

*v*_{i}and throw an identical ball straight down with speed*v*_{i}. Ignoring air resistance, how do the speeds of the balls compare just before they hit the ground?You throw a ball up with initial speed

*v*_{i}and when it reaches its high point at height*h,*you throw another ball into the air with the same initial speed*v*_{i}. Will the two balls cross at half the height*h,*or more than half, or less than half? Explain.When a coin is tossed directly upward, what can you say about its velocity and acceleration at the high point of the toss?

The net force acting on an object is constant. Under what circumstances does the object move along a straight line? Under what circumstances does the object move along a curved path?

You decide to test your physics knowledge while going over a waterfall in a barrel. You take a baseball into the barrel with you and as you are falling vertically downward, you let go of the ball. What do you expect to see for the motion of the ball relative to the barrel? Will the ball fall faster than you and move toward the bottom of the barrel? Will it move slower than you and approach the top of the barrel. Or will it hover apparently motionless within the falling barrel? Explain. [

*Warning:*Do not try this at home.]

Multiple-Choice Questions

A ball is thrown straight up into the air. Ignore air resistance. While the ball is in the air its acceleration

(a) increases.

(b) is zero.

(c) remains constant.

(d) decreases on the way up and increases on the way down.

(e) changes direction.

A leopard starts from rest at

*t*= 0 and runs in a straight line with a constant acceleration until*t*= 3.0 s. The distance covered by the leopard between*t*= 1.0 s and*t*= 2.0 s is(a) the same as the distance covered during the first second.

(b) twice the distance covered during the first second.

(c) three times the distance covered during the first second.

(d) four times the distance covered during the first second.

A kicker kicks a football from the 5-yd line to the 45-yd line (both on the same half of the field). Ignoring air resistance, where along the trajectory is the speed of the football a minimum?

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(a) at the 5-yd line, just after the football leaves the kicker's foot

(b) at the 45-yd line, just before the football hits the ground

(c) at the 15-yd line, while the ball is still going higher

(d) at the 35-yd line, while the ball is coming down

(e) at the 25-yd line, when the ball is at the top of its trajectory

Two balls, identical except for color, are projected horizontally from the roof of a tall building at the same instant. The initial speed of the red ball is twice the initial speed of the blue ball. Ignoring air resistance,

(a) the red ball reaches the ground first.

(b) the blue ball reaches the ground first.

(c) both balls land at the same instant with different speeds.

(d) both balls land at the same instant with the same speed.

A person stands on the roof garden of a tall building with one ball in each hand. If the red ball is thrown horizontally off the roof and the blue ball is simultaneously dropped over the edge, which statement is true?

(a) Both balls hit the ground at the same time, but the red ball has a higher speed just before it strikes the ground.

(b) The blue ball strikes the ground first, but with a lower speed than the red ball.

(c) The red ball strikes the ground first with a higher speed than the blue ball.

(d) Both balls hit the ground at the same time with the same speed.

A ball is thrown into the air and follows a parabolic trajectory. At the highest point in the trajectory,

(a) the velocity is zero, but the acceleration is not zero.

(b) both the velocity and the acceleration are zero.

(c) the acceleration is zero, but the velocity is not zero.

(d) neither the acceleration nor the velocity are zero.

A ball is thrown into the air and follows a parabolic trajectory. Point

*A*is the highest point in the trajectory and point*B*is a point as the ball is falling back to the ground. Choose the correct relationship between the speeds and the magnitudes of the acceleration at the two points.(a)

*v*_{A}>*v*_{B}and*a*_{A}=*a*_{B}(b)

*v*_{A}<*v*_{B}and*a*_{A}>*a*_{B}(c)

*v*_{A}=*v*_{B}and*a*_{A}≠*a*_{B}(d)

*v*_{A}<*v*_{B}and*a*_{A}=*a*_{B}

You are standing on a bathroom scale in an elevator. In which of these situations must the scale read the same as when the elevator is at rest? Explain.

(a) Moving up at constant speed.

(b) Moving up with increasing speed.

(c) In free fall (after the elevator cable has snapped).

A thin string that can support a weight of 35.0 N, but breaks under any larger weight, is attached to the ceiling of an elevator. How large a mass can be attached to the string if the initial acceleration as the elevator starts to ascend is 3.20 m/s

^{2}?(a) 3.57 kg

(b) 2.69 kg

(c) 4.26 kg

(d) 2.96 kg

(e) 5.30 kg

A woman stands on a bathroom scale in an elevator that is not moving. The scale reads 500 N. The elevator then moves downward at a constant velocity of 4.5 m/s. What does the scale read while the elevator descends with constant velocity?

(a) 100 N

(b) 250 N

(c) 450 N

(d) 500 N

(e) 750 N

A 70.0-kg man stands on a bathroom scale in an elevator. What does the scale read if the elevator is slowing down at a rate of 3.00 m/s

^{2}while descending?(a) 70 kg

(b) 476 N

(c) 686 N

(d) 700 N

(e) 896 N

A small plane climbs with a constant velocity of 250 m/s at an angle of 28° with respect to the horizontal. Which statement is true concerning the magnitude of the net force on the plane?

(a) It is equal to zero.

(b) It is equal to the weight of the plane.

(c) It is equal to the magnitude of the force of air resistance.

(d) It is less than the weight of the plane but greater than zero.

(e) It is equal to the component of the weight of the plane in the direction of motion.

Two blocks are connected by a light string passing over a pulley (see the figure). The block with mass

*m*_{1}slides on the frictionless horizontal surface, while the block with mass*m*_{2}hangs vertically (*m*_{1}>*m*_{2}). The tension in the string is(a) zero.

(b) less than

*m*_{2}*g.*(c) equal to

*m*_{2}*g.*(d) greater than

*m*_{2}*g,*but less than*m*_{1}*g.*(e) equal to

*m*_{1}*g.*(f) greater than

*m*_{1}*g.*Tutorial: Pulley 1

Tutorial: Pulley 2

** Questions 14–16.** In Fig. 4.29, two projectiles launched with the same initial speed but at different launch angles 30° and 60° land at the same spot. Ignore air resistance. Answer choices:

(a) Projectile launched at 30°

(b) Projectile launched at 60°

(c) They are equal.

14. Which has the larger horizontal velocity component

*v*_{x}?15. Which has a longer time of flight Δ

*t*(time interval between launch and hitting the ground)?16. For which is the product

*v*_{x}Δ*t*larger?

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Problems

Combination conceptual/quantitative problem

Biological or medical application

Challenging problem

Detailed solution in the Student Solutions Manual

Problems paired by concept

Interactive or tutorial

4.1 Motion Along a Line Due to a Constant Net Force

**1.**An airplane lands and starts down the runway at a southwest velocity of 55 m/s. What constant acceleration allows it to come to a stop in 1.0 km?2. A car is speeding up and has an instantaneous velocity of 1.0 m/s in the +

*x*-direction when a stopwatch reads 10.0 s. It has a constant acceleration of 2.0 m/s^{2}in the +*x*-direction. (a) What change in speed occurs between*t*= 10.0 s and*t*= 12.0 s? (b) What is the speed when the stopwatch reads 12.0 s?3. While passing a slower car on the highway, you accelerate uniformly from 17.4 m/s to 27.3 m/s in a time of 10.0 s. (a) How far do you travel during this time? (b) What is your acceleration magnitude?

4. In a game of shuffleboard, a disk with an initial speed of 3.2 m/s travels 6.0 m before coming to rest. (a) What was the magnitude of the average acceleration of the disk? (b) What was the coefficient of kinetic friction acting on the disk?

**5.**A skier with a mass of 63 kg starts from rest and skis down an icy (frictionless) slope that has a length of 50 m at an angle of 32° with respect to the horizontal. At the bottom of the slope, the path levels out and becomes horizontal, the snow becomes less icy, and the skier begins to slow down, coming to rest in a distance of 140 m along the horizontal path. (a) What is the speed of the skier at the bottom of the slope? (b) What is the coefficient of kinetic friction between the skier and the horizontal surface?6. A cheetah can accelerate from rest to 24 m/s in 2.0 s. Assuming the acceleration is constant over the time interval, (a) what is the magnitude of the acceleration of the cheetah? (b) What is the distance traveled by the cheetah in these 2.0 s? (c) A runner can accelerate from rest to 6.0 m/s in the same time, 2.0 s. What is the magnitude of the acceleration of the runner? By what factor is the cheetah's average acceleration magnitude greater than that of the runner?

7. The forces on a small airplane (mass 1160 kg) in horizontal flight heading eastward are as follows: gravity = 16.000 kN downward, lift = 16.000 kN upward, thrust = 1.800 kN eastward, and drag = 1.400 kN westward. At

*t*= 0, the plane's speed is 60.0 m/s. If the forces remain constant, how far does the plane travel in the next 60.0 s?8. A train of mass 55,200 kg is traveling along a straight, level track at 26.8 m/s (60.0 mi/h). Suddenly the engineer sees a truck stalled on the tracks 184 m ahead. If the maximum possible braking force has magnitude 84.0 kN, can the train be stopped in time?

**9.**You are driving your car along a country road at a speed of 27.0 m/s. As you come over the crest of a hill, you notice a farm tractor 25.0 m ahead of you on the road, moving in the same direction as you at a speed of 10.0 m/s. You immediately slam on your brakes and slow down with a constant acceleration of magnitude 7.00 m/s^{2}. Will you hit the tractor before you stop? How far will you travel before you stop or collide with the tractor? If you stop, how far is the tractor in front of you when you finally stop?10. In a cathode ray tube, electrons are accelerated from rest by a constant electric force of magnitude 6.4 × 10

^{-17}N during the first 2.0 cm of the tube's length; then they move at essentially constant velocity another 45 cm before hitting the screen. (a) Find the speed of the electrons when they hit the screen. (b) How long does it take them to travel the length of the tube?11. A 10.0-kg watermelon and a 7.00-kg pumpkin are attached to each other via a cord that wraps over a pulley, as shown. Friction is negligible everywhere in this system. (a) Find the accelerations of the pumpkin and the watermelon. Specify magnitude and direction. (b) If the system is released from rest, how far along the incline will the pumpkin travel in 0.30 s? (c) What is the speed of the watermelon after 0.20 s?

Tutorial: Pulley 1

Tutorial: Pulley 2

12. Two blocks are connected by a lightweight, flexible cord that passes over a frictionless pulley. If

*m*_{1}= 3.6 kg and*m*_{2}= 9.2 kg, and block 2 is initially at rest 140 cm above the floor, how long does it take block 2 to reach the floor?**13.**While an elevator of mass 832 kg moves downward, the tension in the supporting cable is a constant 7730 N. Between*t*= 0 and*t*= 4.00 s, the elevator's displacement is 5.00 m downward. What is the elevator's speed at*t*= 4.00 s?

4.2 Visualizing Motion Along a Line with Constant Acceleration

14. The graph is of

*v*_{x}versus*t*for an object moving along the*x*-axis. How far does the object move between*t*= 9.0 s and*t*= 13.0 s? Solve using two methods: a graphical analysis and an algebraic solution.- Page 139
**15.**The graph is of*v*_{x}versus*t*for an object moving along the*x*-axis. What is the average acceleration between*t*= 5.0 s and*t*= 9.0 s? Solve using two methods: a graphical analysis and an algebraic solution.

16. A train, traveling at a constant speed of 22 m/s, comes to an incline with a constant slope. While going up the incline, the train slows down with a constant acceleration of magnitude 1.4 m/s

^{2}. (a) Draw a graph of*v*_{x}versus*t*where the*x*-axis points up the incline. (b) What is the speed of the train after 8.0 s on the incline? (c) How far has the train traveled up the incline after 8.0 s? (d) Draw a motion diagram, showing the train's position at 2.0-s intervals.**17.**The St. Charles streetcar in New Orleans starts from rest and has a constant acceleration of 1.20 m/s^{2}for 12.0 s. (a) Draw a graph of*v*_{x}versus*t.*(b) How far has the train traveled at the end of the 12.0 s? (c) What is the speed of the train at the end of the 12.0 s? (d) Draw a motion diagram, showing the trolley car's position at 2.0-s intervals.18. A streetcar named Desire travels between two stations 0.60 km apart. Leaving the first station, it accelerates for 10.0 s at 1.0 m/s

^{2}and then travels at a constant speed until it is near the second station, when it brakes at 2.0 m/s^{2}in order to stop at the station. How long did this trip take? [*Hint:*What's the average velocity?]

19. In the physics laboratory, a glider is released from rest on a frictionless air track inclined at an angle. If the glider has gained a speed of 25.0 cm/s in traveling 50.0 cm from the starting point, what was the angle of inclination of the track? Draw a graph of

*v*_{x}(*t*) when the positive*x*-axis points down the track.20. A 10.0-kg block is released from rest on a frictionless track inclined at an angle of 55°. (a) What is the net force on the block after it is released? (b) What is the acceleration of the block? (c) If the block is released from rest, how long will it take for the block to attain a speed of 10.0 m/s? (d) Draw a motion diagram for the block. (e) Draw a graph of

*v*_{x}(*t*) for values of velocity between 0 and 10 m/s. Let the positive*x*-axis point down the track.

**21.**A 6.0-kg block, starting from rest, slides down a frictionless incline of length 2.0 m. When it arrives at the bottom of the incline, its speed is*v*_{f}. At what distance from the top of the incline is the speed of the block 0.50*v*_{f}?22. A train is traveling south at 24.0 m/s when the brakes are applied. It slows down with a constant acceleration to a speed of 6.00 m/s in a time of 9.00 s. (a) Draw a graph of

*v*_{x}versus*t*for a 12-s interval (starting 2 s before the brakes are applied and ending 1 s after the brakes are released). Let the*x*-axis point to the north. (b) What is the acceleration of the train during the 9.00-s interval? (c) How far does the train travel during the 9.00 s?

4.3 Free Fall

23. Grant Hill jumps 1.3 m straight up into the air to slam-dunk a basketball into the net. With what speed did he leave the floor?

24. A penny is dropped from the observation deck of the Empire State building (369 m above ground). With what velocity does it strike the ground? Ignore air resistance.

**25.**(a) How long does it take for a golf ball to fall from rest for a distance of 12.0 m? (b) How far would the ball fall in twice that time?

26. A student, looking toward his fourth-floor dormitory window, sees a flowerpot with nasturtiums (originally on a window sill above) pass his 2.0-m high window in 0.093 s. The distance between floors in the dormitory is 4.0 m. From a window on which floor did the flowerpot fall?

**27.**A balloonist, riding in the basket of a hot air balloon that is rising vertically with a constant velocity of 10.0 m/s, releases a sandbag when the balloon is 40.8 m above the ground. Ignoring air resistance, what is the bag's speed when it hits the ground?28. A 55-kg lead ball is dropped from the leaning tower of Pisa. The tower is 55 m high. (a) How far does the ball fall in the first 3.0 s of flight? (b) What is the speed of the ball after it has traveled 2.5 m downward? (c) What is the speed of the ball 3.0 s after it is released?

**29.**Superman is standing 120 m horizontally away from Lois Lane. A villain drops a rock from 4.0 m directly above Lois. (a) If Superman is to intervene and catch the rock just before it hits Lois, what should be his minimum constant acceleration? (b) How fast will Superman be traveling when he reaches Lois?30. During a walk on the Moon, an astronaut accidentally drops his camera over a 20.0-m cliff. It leaves his hands with zero speed, and after 2.0 s it has attained a velocity of 3.3 m/s downward. How far has the camera fallen after 4.0 s?

31. Glenda drops a coin from ear level down a wishing well. The coin falls a distance of 7.00 m before it strikes the water. If the speed of sound is 343 m/s, how long after Glenda releases the coin will she hear a splash?

32. A stone is launched straight up by a slingshot. Its initial speed is 19.6 m/s and the stone is 1.50 m above the ground when launched. (a) How high above the ground does the stone rise? (b) How much time elapses before the stone hits the ground?

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**33.**A model rocket is fired vertically from rest. It has a net acceleration of 17.5 m/s^{2}. After 1.5 s, its fuel is exhausted and its only acceleration is that due to gravity. (a) Ignoring air resistance, how high does the rocket travel? (b) How long after liftoff does the rocket return to the ground? 34. The model rocket in Problem 33 has a mass of 87 g and you may assume the mass of the fuel is much less than 87 g. (a) What was the net force on the rocket during the first 1.5 s after liftoff? (b) What force was exerted on the rocket by the burning fuel? (c) What was the net force on the rocket after its fuel was spent? (d) The rocket's vertical velocity was zero instantaneously when it was at the top of its trajectory. What were the net force and acceleration on the rocket at this instant?

35. You drop a stone into a deep well and hear it hit the bottom 3.20 s later. This is the time it takes for the stone to fall to the bottom of the well, plus the time it takes for the sound of the stone hitting the bottom to reach you. Sound travels about 343 m/s in air. How deep is the well?

36. A baseball is thrown horizontally from a height of 9.60 m above the ground with a speed of 30.0 m/s. Where is the ball after 1.40 s have elapsed?

**37.**A clump of soft clay is thrown horizontally from 8.50 m above the ground with a speed of 20.0 m/s. Where is the clay after 1.50 s? Assume it sticks in place when it hits the ground.38. A tennis ball is thrown horizontally from an elevation of 14.0 m above the ground with a speed of 20.0 m/s. (a) Where is the ball after 1.60 s? (b) If the ball is still in the air, how long before it hits the ground and where will it be with respect to the starting point once it lands?

4.4 Motion of Projectiles

39. A ball is thrown from a point 1.0 m above the ground. The initial velocity is 19.6 m/s at an angle of 30.0° above the horizontal. (a) Find the maximum height of the ball above the ground. (b) Calculate the speed of the ball at the highest point in the trajectory.

40. An arrow is shot into the air at an angle of 60.0° above the horizontal with a speed of 20.0 m/s. (a) What are the

*x*- and*y*-components of the velocity of the arrow 3.0 s after it leaves the bowstring? (b) What are the*x*- and*y*-components of the displacement of the arrow during the 3.0-s interval?**41.**You are working as a consultant on a video game designing a bomb site for a World War I airplane. In this game, the plane you are flying is traveling horizontally at 40.0 m/s at an altitude of 125 m when it drops a bomb. (a) Determine how far horizontally from the target you should release the bomb. (b) What direction is the bomb moving just before it hits the target?42. A ballplayer standing at home plate hits a baseball that is caught by another player at the same height above the ground from which it was hit. The ball is hit with an initial velocity of 22.0 m/s at an angle of 60.0° above the horizontal. (a) How high will the ball rise? (b) How much time will elapse from the time the ball leaves the bat until it reaches the fielder? (c) At what distance from home plate will the fielder be when he catches the ball?

Tutorial: Projectile 1

Tutorial: Projectile 2

43. You are planning a stunt to be used in an ice skating show. For this stunt a skater will skate down a frictionless ice ramp that is inclined at an angle of 15.0° above the horizontal. At the bottom of the ramp, there is a short horizontal section that ends in an abrupt drop off. The skater is supposed to start from rest somewhere on the ramp, then skate off the horizontal section and fly through the air a horizontal distance of 7.00 m while falling vertically for 3.00 m, before landing smoothly on the ice. How far up the ramp should the skater start this stunt?

44. A suspension bridge is 60.0 m above the level base of a gorge. A stone is thrown or dropped from the bridge. Ignore air resistance. At the location of the bridge

*g*has been measured to be 9.83 m/s^{2}. (a) If you drop the stone, how long does it take for it to fall to the base of the gorge? (b) If you*throw*the stone straight down with a speed of 20.0 m/s, how long before it hits the ground? (c) If you throw the stone with a velocity of 20.0 m/s at 30.0° above the horizontal, how far from the point directly below the bridge will it hit the level ground?**45.**From the edge of the rooftop of a building, a boy throws a stone at an angle 25.0° above the horizontal. The stone hits the ground 4.20 s later, 105 m away from the base of the building. (Ignore air resistance.) (a) For the stone's path through the air, sketch graphs of*x,**y,**v*_{x}, and*v*_{y}as functions of time. These need to be only*qualitatively*correct—you need not put numbers on the axes. (b) Find the initial velocity of the stone. (c) Find the initial height*h*from which the stone was thrown. (d) Find the maximum height*H*reached by the stone.46. Jason is practicing his tennis stroke by hitting balls against a wall. The ball leaves his racquet at a height of 60 cm above the ground at an angle of 80° with respect to the

*vertical.*(a) The speed of the ball as it leaves the racquet is 20 m/s and it must travel a distance of 10 m before it reaches the wall. How far above the ground does the ball strike the wall? (b) Is the ball on its way up or down when it hits the wall?47. You have been employed by the local circus to plan their human cannonball performance. For this act, a spring-loaded cannon will shoot a human projectile, the Great Flyinski, across the big top to a net below. The net is located 5.0 m lower than the muzzle of the cannon from which the Great Flyinski is launched. The cannon will shoot the Great Flyinski at an angle of 35.0° above the horizontal and at a speed of 18.0 m/s. The ringmaster has asked that you decide how far from the cannon to place the net so that the Great Flyinski will land in the net and not be splattered on the floor, which would greatly disturb the audience. What do you tell the ringmaster?

Interactive: Projectile Motion

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48. A circus performer is shot out of a cannon and flies over a net that is placed horizontally 6.0 m from the cannon. When the cannon is aimed at an angle of 40° above the horizontal, the performer is moving in the horizontal direction and just barely clears the net as he passes over it. What is the muzzle speed of the cannon and how high is the net?

**49.**A cannonball is catapulted toward a castle. The cannonball's velocity when it leaves the catapult is 40 m/s at an angle of 37° with respect to the horizontal and the cannonball is 7.0 m above the ground at this time. (a) What is the maximum height above the ground reached by the cannonball? (b) Assuming the cannonball makes it over the castle walls and lands back down on the ground, at what horizontal distance from its release point will it land? (c) What are the*x*- and*y*-components of the cannonball's velocity just before it lands? The*y*-axis points up.50. After being assaulted by flying cannonballs, the knights on the castle walls (12 m above the ground) respond by propelling flaming pitch balls at their assailants. One ball lands on the ground at a distance of 50 m from the castle walls. If it was launched at an angle of 53° above the horizontal, what was its initial speed?

51. The citizens of Paris were terrified during World War I when they were suddenly bombarded with shells fired from a long-range gun known as Big Bertha. The barrel of the gun was 36.6 m long and it had a muzzle speed of 1.46 km/s. When the gun's angle of elevation was set to 55°, what would be the range? For the purposes of solving this problem, ignore air resistance. (The actual range at this elevation was 121 km; air resistance cannot be ignored for the high muzzle speed of the shells.)

52. The range

*R*of a projectile is defined as the magnitude of the horizontal displacement of the projectile*when it returns to its original altitude.*In other words, the range is the distance between the launch point and the impact point on flat ground. A projectile is launched at*t*= 0 with initial speed*v*_{i}at an angle θ above the horizontal. (a) Find the time*t*at which the projectile returns to its original altitude. (b) Show that the range is[

*Hint:*Use the trigonometric identity sin 2 θ = 2 sin θ cos θ.]**53.**Use the expression in Problem 52 to find (a) the maximum range of a projectile with launch speed*v*_{i}and (b) the launch angle θ at which the maximum range occurs.54. A projectile is launched at

*t*= 0 with initial speed*v*_{i}at an angle θ above the horizontal. (a) What are*v*_{x}and*v*_{y}at the projectile's highest point? (b) Find the time*t*at which the projectile reaches its maximum height. (c) Show that the maximum height*H*of the projectile is55. Two angles are complementary when their sum is 90.0°. Find the ranges for two projectiles launched with identical initial speeds of 36.2 m/s at angles of elevation above the horizontal that are complementary pairs. (a) For one trial, the angles of elevation are 36.0° and 54.0°. (b) For the second trial, the angles of elevation are 23.0° and 67.0°. (c) Finally, the angles of elevation are both set to 45.0°. (d) What do you notice about the range values for each complementary pair of angles? At which of these angles was the range greatest?

4.5 Apparent Weight

56. Oliver has a mass of 76.2 kg. He is riding in an elevator that has a downward acceleration of 1.37 m/s

^{2}. With what magnitude force does the elevator floor push upward on Oliver?**57.**Yolanda, whose mass is 64.2 kg, is riding in an elevator that has an upward acceleration of 2.13 m/s^{2}. What force does she exert on the floor of the elevator?58. When on the ground, Ian's weight is measured to be 640 N. When Ian is on an elevator, his apparent weight is 700 N. What is the net force on the system (Ian and the elevator) if their combined mass is 1050 kg?

59. While on an elevator, Jaden's apparent weight is 550 N. When he is on the ground, the scale reading is 600 N. What is Jaden's acceleration?

60. You are standing on a bathroom scale inside an elevator. Your weight is 140 lb, but the reading of the scale is 120 lb. (a) What is the magnitude and direction of the acceleration of the elevator? (b) Can you tell whether the elevator is speeding up or slowing down?

**61.**Refer to Example 4.12. What is the apparent weight of the same passenger (weighing 598 N) in the following situations? In each case, the magnitude of the elevator's acceleration is 0.50 m/s^{2}. (a) After having stopped at the fifteenth floor, the passenger pushes the eighth floor button; the elevator is beginning to move downward. (b) The elevator is moving downward and is slowing down as it nears the eighth floor.

62. Luke stands on a scale in an elevator that has a constant acceleration upward. The scale reads 0.960 kN. When Luke picks up a box of mass 20.0 kg, the scale reads 1.200 kN. (The acceleration remains the same.) (a) Find the acceleration of the elevator. (b) Find Luke's weight.

**63.**Felipe is going for a physical before joining the swim team. He is concerned about his weight, so he carries his scale into the elevator to check his weight while heading to the doctor's office on the twenty-first floor of the building. If his scale reads 750 N while the elevator has an upward acceleration of 2.0 m/s^{2}, what does the nurse measure his weight to be?

Page 142

Comprehensive Problems

64. A drag racer crosses the finish line of a 400.0-m track with a final speed of 104 m/s. (a) Assuming constant acceleration during the race, find the racer's time and the minimum coefficient of static friction between the tires and the road. (b) If, because of bad tires or wet pavement, the acceleration were 30.0% smaller, how long would it take to finish the race?

**65.**An African swallow carrying a very small coconut is flying horizontally with a speed of 18 m/s. (a) If it drops the coconut from a height of 100 m above the Earth, how long will it take before the coconut strikes the ground? (b) At what horizontal distance from the release point will the coconut strike the ground?

66. A ball is thrown horizontally off the edge of a cliff with an initial speed of 20.0 m/s. (a) How long does it take for the ball to fall to the ground 20.0 m below? (b) How long would it take for the ball to reach the ground if it were dropped from rest off the cliff edge? (c) How long would it take the ball to fall to the ground if it were thrown at an initial velocity of 20.0 m/s but 18° below the horizontal?

**67.**A marble is rolled so that it is projected horizontally off the top landing of a staircase. The initial speed of the marble is 3.0 m/s. Each step is 0.18 m high and 0.30 m wide. Which step does the marble strike first?

68. You are serving as a consultant for the newest James Bond film. In one scene, Bond must fire a projectile from a cannon and hit the enemy headquarters located on the top of a cliff 75.0 m above and 350 m from the cannon. The cannon will shoot the projectile at an angle of 40.0° above the horizontal. The director wants to know what the speed of the projectile must be when it is fired from the cannon so that it will hit the enemy headquarters. What do you tell him? [

*Hint:*Don't assume the projectile will hit the headquarters at the highest point of its flight.]**69.**An airplane is traveling from New York to Paris, a distance of 5.80 × 10^{3}km. Ignore the curvature of the Earth. (a) If the cruising speed of the airplane is 350.0 km/h, how much time will it take for the airplane to make the round-trip on a calm day? (b) If a steady wind blows from New York to Paris at 60.0 km/h, how much time will the round-trip take? (c) How much time will it take if there is a crosswind of 60.0 km/h?70. In free fall, we assume the acceleration to be constant. Not only is air resistance ignored, but the gravitational field strength is assumed to be constant. From what height can an object fall to the Earth's surface such that the gravitational field strength changes less than 1.000% during the fall?

71. The graph for this problem shows the vertical velocity component

*v*_{y}of a bouncing ball as a function of time. The*y*-axis points up. Answer these questions based on the data in the graph. (a) At what time does the ball reach its maximum height? (b) For how long is the ball in contact with the floor? (c) What is the maximum height of the ball? (d) What is the acceleration of the ball while in the air? (e) What is the average acceleration of the ball while in contact with the floor?72. A rocket engine can accelerate a rocket launched from rest vertically up with an acceleration of 20.0 m/s

^{2}. However, after 50.0 s of flight the engine fails. (a) What is the rocket's altitude when the engine fails? (b) When does it reach its maximum height? (c) What is the maximum height reached? [*Hint:*A graphical solution may be easiest.] (d) What is the velocity of the rocket just before it hits the ground?**73.**A particle has a constant acceleration of 5.0 m/s^{2}to the east. At time*t*= 0, it is 2.0 m east of the origin and its velocity is 20 m/s north. What are the components of its position vector at*t*= 2.0 s?74. A baseball batter hits a long fly ball that rises to a height of 44 m. An outfielder on the opposing team can run at 7.6 m/s. What is the farthest the fielder can be from where the ball will land so that it is possible for him to catch the ball?

75. A 15-kg crate starts at rest at the top of a 60.0° incline. The coefficients of friction are μ

_{s}= 0.40 and μ_{k}= 0.30. The crate is connected to a hanging 8.0-kg box by an ideal rope and pulley. (a) As the crate slides down the incline, what is the tension in the rope? (b) How long does it take the crate to slide 2.00 m down the incline? (c) To push the crate back up the incline at constant speed, with what force*P*should Pauline push on the crate (parallel to the incline)? (d) What is the smallest mass that you could substitute for the 8.0-kg box to keep the crate from sliding down the incline?76. A beanbag is thrown horizontally from a dorm room window a height

*h*above the ground. It hits the ground a horizontal distance*h*(the*same*distance*h*) from the dorm directly below the window from which it was thrown. Ignoring air resistance, find the direction of the beanbag's velocity just before impact.- Page 143
**77.**An unmarked police car starts from rest just as a speeding car passes at a speed of*v.*If the police car speeds up with a constant acceleration of*a,*what is the speed of the police car when it catches up to the speeder, who does not realize she is being pursued and does not vary her speed? 78. A seagull is flying horizontally 8.00 m above the ground at 6.00 m/s. The bird is carrying a clam in its beak and plans to crack the clamshell by dropping it on some rocks below. Ignoring air resistance, (a) what is the horizontal distance to the rocks at the moment that the seagull should let go of the clam? (b) With what speed relative to the rocks does the clam smash into the rocks? (c) With what speed relative to the seagull does the clam smash into the rocks?

79. The minimum stopping distance of a car moving at 30.0 mi/h is 12 m. Under the same conditions (so that the maximum braking force is the same), what is the minimum stopping distance for 60.0 mi/h? Work by proportions to avoid converting units.

80. A car traveling at 29 m/s (65 mi/h) runs into a bridge abutment after the driver falls asleep at the wheel. (a) If the driver is wearing a seat belt and comes to rest within a 1.0-m distance, what is his acceleration (assumed constant)? (b) A passenger who isn't wearing a seat belt is thrown into the windshield and comes to a stop in a distance of 10.0 cm. What is the acceleration of the passenger?

**81.**Find the point of no return for an airport runway of 1.50 mi in length if a jet plane can accelerate at 10.0 ft/s^{2}and decelerate at 7.00 ft/s^{2}. The point of no return occurs when the pilot can no longer abort the takeoff without running out of runway. What length of time is available from the start of the motion in which to decide on a course of action?82. A helicopter is flying horizontally at 8.0 m/s and an altitude of 18 m when a package of emergency medical supplies is ejected horizontally backward with a speed of 12 m/s

*relative to the helicopter.*Ignoring air resistance, what is the horizontal distance between the package and the helicopter when the package hits the ground?83. A locust jumps at an angle of 55.0° and lands 0.800 m from where it jumped. (a) What is the maximum height of the locust during its jump? Ignore air resistance. (b) If it jumps with the same initial speed at an angle of 45.0°, would the maximum height be larger or smaller? (c) What about the range? (d) Calculate the maximum height and range for this angle.

84. A toboggan is sliding down a snowy slope. The table shows the speed of the toboggan at various times during its trip. (a) Make a graph of the speed as a function of time. (b) Judging by the graph, is it plausible that the toboggan's acceleration is constant? If so, what is the acceleration? (c) Ignoring friction, what is the angle of incline of the slope? (d) If friction is significant, is the angle of incline larger or smaller than that found in (c)? Explain.

**Time Elapsed,***t***(s)****Speed of Toboggan,***v***(m/s)**0

0

1.14

2.8

1.62

3.9

2.29

5.6

2.80

6.8

**85.**Show that for a projectile launched at an angle of 45° the maximum height of the projectile is one quarter of the range (the distance traveled on flat ground).86. An airtrack glider, 8.0 cm long, blocks light as it goes through a photocell gate. The glider is released from rest on a frictionless inclined track and the gate is positioned so that the glider has traveled 96 cm when it is in the middle of the gate. The timer gives a reading of 333 ms for the glider to pass through this gate. Friction is negligible. What is the acceleration (assumed constant) of the glider along the track?

87. You want to make a plot of the trajectory of a projectile. That is, you want to make a plot of the height

*y*of the projectile as a function of horizontal distance*x.*The projectile is launched from the origin with an initial speed*v*_{i}at an angle θ above the horizontal. Show that the equation of the trajectory followed by the projectile is88. Locusts can jump to heights of 0.30 m. (a) Assuming the locust jumps straight up, and ignoring air resistance, what is the takeoff speed of the locust? (b) The locust actually jumps at an angle of about 55° to the horizontal, and air resistance is not negligible. The result is that the takeoff speed is about 40% higher than the value you calculated in part (a). If the mass of the locust is 2.0 g and its body moves 4.0 cm in a straight line while accelerating from rest to the takeoff speed, calculate the acceleration of the locust (assumed constant). (c) Ignore the locust's weight and estimate the force exerted on the hind legs by the ground. Compare this force with the locust's weight. Was it reasonable to ignore the locust's weight?

- Page 144
**89.**In Fig. 4.3, the block of mass*m*_{1}slides to the right with coefficient of kinetic friction μ_{k}on a horizontal surface. The block is connected to a hanging block of mass*m*_{2}by a light cord that passes over a light, frictionless pulley. (a) Find the acceleration of each of the blocks and the tension in the cord. (b) Check your answers in the special cases*m*_{1}< <*m*_{2},*m*_{1}> >*m*_{2}, and*m*_{1}=*m*_{2}. (c) For what value of*m*_{2}(if any) do the two blocks slide at constant velocity? What is the tension in the cord in that case? 90. When salmon head upstream to spawn, they often must make their way up a waterfall. If the water is not moving too fast, the salmon can swim right up through the falling water. If the water is falling with too great a speed, the salmon jump out of the water to get to a place in the waterfall where the water is not falling so fast. When humans build dams that interrupt the usual route followed by the salmon, artificial fish ladders must also be built to allow the salmon to get back uphill to the spawning area. These fish ladders consist of a series of small waterfalls with still pools of water in between them (see the figure). Assume that the water is at rest in the pools at the top and bottom of one “rung” of the fish ladder, that water falls straight down from one pool to the next, and that salmon can swim at 5.0 m/s with respect to the water. (a) What is the maximum height of a waterfall up which the salmon can swim without having to jump? (b) If a waterfall is 1.5 m high, how high must the salmon jump to get to water through which it can swim? Assume that they jump straight up. (c) What initial speed must a salmon have to jump the height found in part (b)? (d) For a 1.0-m-high waterfall, how fast will the salmon be swimming with respect to the ground when it starts swimming up the waterfall?

91. Your current case as an FBI investigator involves a possible assassination attempt. The crime scene is a tall building, about 150 m high. As a foreign official was walking into the building, a flowerpot fell from above him and nearly landed on his head. One witness says that someone accidentally knocked the flowerpot from a twenty-fourth-story window, 94 m above the ground. Another witness was inside the building looking out of an eighteenth-story window when she saw the flowerpot fall. She claims that the flowerpot took exactly 0.044 s to fall from the top of the window to the bottom of the window, a distance of 1.5 m. (She knows this because she was videotaping her pet dachshund doing tricks and the flowerpot falling past the window was filmed in the background.) The top of the eighteenth-story window is 75 m above the ground. Could the flowerpot have fallen with zero initial velocity from the twenty-fourth-story window?

92. Jesse James and his gang are planning to rob the train carrying the payroll from the Lost Gulch mine. A Wells Fargo guard on the train spots Jesse astride his horse at a perpendicular distance of 0.300 km from the train tracks. The train is moving at 25.0 m/s on a straight track. The guard wants to frighten Jesse and the desperadoes away by shooting a hole in Jesse's ten-gallon hat, which happens to be at the same height as the guard's gun (see the figure). The muzzle velocity of the guard's gun is 0.350 km/s. Ignore air resistance. (a) Should he aim the gun directly at Jesse's hat, in front of the hat, or behind the hat? Explain. (b) Unsure of the correct angle to aim the gun, the guard decides to aim his gun at a right angle to the track and fire the bullet a little before the time when he will be directly opposite to Jesse. At what distance before the point where the guard is directly opposite Jesse James should the guard fire? (c) Does the guard need to worry about the force of gravity on the bullet? If so, how far above the hat should he aim?

Answers to Practice Problems

**4.1**0.31 m to the right (block 1) and 0.31 m down (block 2)**4.2**Impossible to pull the crate up with a single pulley. The entire weight of the crate would be supported by a single strand of cable and that weight exceeds the breaking strength of the cable.- Page 145
**4.3**20 s **4.4**2500 N**4.5**5.00 s after they leave the starting point; 4.00 km/s in the +*x*-direction**4.6**(a) down the incline; (b) up the incline; (c) 0.2 m/s^{2}down the incline**4.7**(a) 3.8 m; (b) 3.00 s**4.8**2**4.9****4.10**Ignoring air resistance, the two arrows have the same constant horizontal velocity component:*v*_{2x}−*v*_{1x}= 0 (choosing the*x*-axis horizontal and the*y*-axis up). Their vertical velocity components are different, but they*change at the same rate,*so*v*_{2y}−*v*_{1y}stays constant. The difference in their velocities () stays constant. This constant difference in their velocities makes the difference in their positions () change with time.**4.11***v*_{fx}= 500.0 m/s;*v*_{fy}= −19.8 m/s; bullet enters the water at an angle of 2.27° below the horizontal.**4.12**(a) 392 N;

(b) 431 N

Answers to Checkpoints

**4.1**Only if the plane's acceleration is constant must its average velocity be 470 km/h west. If its acceleration is not constant, the average velocity is not necessarily 470 km/h west. To find the average velocity, we would divide the plane's displacement by the time interval.**4.2**Yes. For motion along an incline, it simplifies the problem to choose one axis parallel to the incline and the other perpendicular to the incline.**4.3**Yes. If you throw a ball upward, it is in free fall as soon as it loses contact with your hand.**4.4**The horizontal velocity component does not change, while the vertical component is zero at the highest point, so the velocity is directed horizontally. The acceleration is constant and directed vertically downward throughout the flight, including at the highest point.**4.5**Your velocity is downward and decreasing in magnitude, so your acceleration is upward. Then the upward normal force exerted on you by the scale must be greater than your weight. The scale reading is greater than your weight.