According to Eq. 7–4, the longer the impact time of an impulse, the smaller the force

What is the magnitude of the momentum of a 28-g sparrow flying with a speed of 8.4 m/s?

A constant friction force of 25 N acts on a 65-kg skier for 20 s. What is the skiers change in velocity?

A 0.145-kg baseball pitched at 39.0 m/s is hit on a horizontal line drive straight back toward the pitcher at 52.0 m/s. If the contact time between bat and ball is 3.0 x 10_3$. calculate the average force between the ball and bat during contact.

A child in a boat throws a 6.40-kg package out horizontally with a speed of 10.0 m/s. Fig. 7-3 Calculate the velocity of the boat immediately after, assuming it was initially at rest. The mass of the child is 26.0 kg. and that of the boat is 45.0 kg. Ignore water resistance.

Calculate the force exerted on a rocket, given that the propelling gases are expelled at a rate of I500kg/s with a speed of 4.0 X 104m/s (at the moment of takeoff).

A 95-kg halfback moving at 4.1 m/s on an apparent breakaway for a touchdown is tackled from behind. When he was tackled by an 85-kg cornerback running at 5.5 m/s in the same direction, what was their mutual speed immediately after the tackle?

A 12,600-kg railroad car travels alone on a level frictionless track with a constant speed of 18.0 m/s. A 5350-kg load, initially at rest, is dropped onto the car. What will be the cars new speed?

A 9300-kg boxcar traveling at 0 m/s strikes a second boxcar at rest. The two stick together and move off with a speed of 6.0 m/s. What is the mass of the second car?

During a Chicago storm, winds can whip horizontally at speeds of 100 km/h. If the air strikes a person at the rate of 40 kg/s per square meter and is brought to rest, estimate the force of the wind on a person. Assume the person is 50 m high and 0.50 m wide. Compare to the typical maximum force of friction (/x = 0) between the person and the ground, if the person has a mass of 70 kg.

A 3800-kg open railroad car coasts along with a constant speed of 8.60 m/s on a level track. Snow begins to fall vertically and fills the car at a rate of 3.50 kg/min. Ignoring friction with the tracks, what is the speed of the car after 90.0 min?

An atomic nucleus initially moving at 420m/$ emits an alpha particle in the direction of its velocity, and the remaining nucleus slows to 350 m/s. If the alpha particle has a mass of 4.0 u and the original nucleus has a mass of 222 u. what speed does the alpha particle have when it is emitted?

A 23-g bullet traveling 230 m/s penetrates a 2.0-kg block of wood and emerges cleanly at 170 m/s. If the block is stationary on a frictionless surface when hit. how fast does it move after the bullet emerges?

A 975-kg two-stage rocket is traveling at a speed of 5.80 x 103 m/s with respect to Earth when a pre- designed explosion separates the rocket into two sections of equal mass that then move at a speed of 2.20 x 103 m/s relative to each other along the original line of motion, (a) What arc the speed and direction of each section (relative to Earth) after the explosion? (b) How much energy was supplied by the explosion? fHint: What is the change in ke as a result of the explosion?)

A rocket of total mass 3180 kg is traveling in outer space with a velocity of 115 m/s. To alter its course by 35.0, its rockets can be fired briefly in a direction perpendicular to its original motion. If the rocket gases are expelled at a speed of 1750 m/s. how much mass must be expelled?

A golf ball of mass 0.045 kg is hit off the tee at a speed of 45 m/s. The golf club was in contact with the ball for 3.5 x 10-3 S- Find (a) the impulse imparted to the golf ball, and (b) the average force exerted on the ball by the golf club.

A 12-kg hammer strikes a nail at a velocity of 8.5 m/s and comes to rest in a time interval of 8.0 ms. (a) What is the impulse given to the nail? (b) What is the average force acting on the nail?

A tennis ball of mass m = 0.060 kg and speed v = 25 m/s strikes a wall at a 45 angle and rebounds with the same speed at 45 (Fig. 7-32). What is the impulse (magnitude and direction) given to the ball?

You are the design engineer in charge of the crashworthiness of new automobile models. Cars are tested by smashing them into fixed, massive barriers at 50 km/h (30mph). A new model of mass 1500 kg takes 0.15 s from the time of impact until it is brought to rest, (a) Calculate the average force exerted on the car by the barrier. (b) Calculate the average deceleration of the car.

A 95-kg fullback is running at 4.0 m/s to the east and is stopped in 0.75 s by a head-on tackle by a tackier running due west. Calculate (a) the original momentum of the fullback, (/>) the impulse exerted on the fullback, (c) the impulse exerted on the tackier, and (d) the average force exerted on the tackier.

Suppose the force acting on a tennis ball (mass 0.060 kg) points in the +x direction and is given by the graph of Fig. 7-33 as a function of time. Use graphical methods to estimate (<?) the total impulse given the ball, and (b) the velocity of the ball after being struck, assuming the ball is being served so it is nearly at rest initially.

From what maximum height can a 75-kg person jump without breaking the lower leg bone of either leg? Ignore air resistance and assume the cm of the person moves a distance of 0.60 m from the standing to the seated position (that is, in breaking the fall). Assume the breaking strength (force per unit area) of bone is 170 X 106N/m2, and its smallest cross-sectional area is 2.5 x 10-4m2. [Hint: Do not try this experimentally.]

A ball of mass 0.440 kg moving cast (+.y direction) with a speed of 3.30 m/s collides head-on with a 0.220-kg ball at rest. If the collision is perfectly elastic, what will be the speed and direction of each ball after the collision?

A 0.450-kg ice puck, moving east with a speed of 3.0 m/s. has a head-on collision with a 0.900-kg puck initially at rest. Assuming a perfectly elastic collision, what will be the speed and direction of each object after the collision?

Two billiard balls of equal mass undergo a perfectly elastic head-on collision. If one ball's initial speed was 2.0 m/s. and the other's was 3.00 m/s in the opposite direction, what will be their speeds after the collision?

A 0.060-kg tennis ball, moving with a speed of 2.50m/s. collides head-on with a 0.090-kg ball initially moving away from it at a speed of 15 m/s. Assuming a perfectly elastic collision, what are the speed and direction of each ball after the collision?

A softball of mass 0.220 kg that is moving with a speed of 8.5 m/s collides head-on and elastically writh another ball initially at rest. Afterward the incoming soft- ball bounces backward with a speed of 3.7 m/s. Calculate (<?) the velocity of the target ball after the collision, and (b) the mass of the target ball.

Two bumper cars in an amusement park ride collide elastically as one approaches the other directly from the rear (Fig. 7-34). Car A has a mass of 450 kg and car B 550 kg, owing to differences in passenger mass. If car A approaches at 4.50 m/s and car B is moving at 3.70 m/s, calculate (a) their velocities after the collision, and (b) the change in momentum of each.

A 0.280-kg croquet ball makes an elastic head-on collision with a second ball initially at rest.Hie second ball moves off with half the original speed of the first ball, (a) What is the mass of the second ball? (b) What fraction of the original kinetic energy ( AKE/KE) gets transferred to the second ball?

In a physics lab. a cube slides down a frictionless incline as shown in Fig. 7-35. and elastically strikes another cube at the bottom that is only one-half its mass. If the incline is 30 cm high and the table is 90 cm off the floor. wrhere does each cube land ? [Hint: Both leave the incline moving horizontally.)

Take the general case of an object of mass mA and velocity uA elastically striking a stationary (vR = 0) object of mass mB head-on. (a) Showr that the final velocities v'A and v'n arc given by (image) What happens in the extreme case wrhen mA is much smaller than wB? Cite a common example of this, (c) What happens in the extreme case when mA is much larger than mB ? Cite a common example of this. (d) What happens in the case w-hen mA = wB ? Cite a common example.

In a ballistic pendulum experiment, projectile 1 results in a maximum height It of the pendulum equal to 2.6 cm. A second projectile causes the the pendulum to swing twice as high, h2 = 5.2 cm. The second projectile was how many times faster than the first?

A 28-g rifle bullet traveling 230m/s buries itself in a 3.6-kg pendulum hanging on a 2.8-m-long string, which makes the pendulum swing upward in an arc. Determine the vertical and horizontal components of the pendulum's displacement.

Derive a formula for the fraction of kinetic energy lost, Ake/ke, for the ballistic pendulum collision of Example 710. (b) Evaluate for m = 14.0 g and M = 380 g.

An internal explosion breaks an object, initially at rest, into two pieces, one of which has 5 times the mass of the other. If 7500 J were released in the explosion, how much kinetic energy did each piece acquire?

A 920-kg sports car collides into the rear end of a 2300-kg SUV stopped at a red light. The bumpers lock, the brakes are locked, and the two cars skid forward 2.8 m before stopping. The police officer, knowing that the coefficient of kinetic friction between tires and road is 0.80. calculates the speed of the sports car at impact. What was that speed?

A ball is dropped from a height of 50 m and rebounds to a height of 20 m. Approximately how many rebounds will the ball make before losing 90% of its energy?

A measure of inelasticity in a head-on collision of tw'o objects is the coefficient of restitution, e. defined as V'A ~ e VB ~ ^ A where vA - *4 is the relative velocity of the two objects after the collision and - vA is their relative velocity before it. (a) Show' that e = 1 for a perfectly elastic collision. and e = 0 for a completely inelastic collision. (6) A simple method for measuring the coefficient of restitution for an object colliding with a very hard surface like steel is to drop the object onto a heavy steel plate, as shown in Fig. 7-36. Determine a formula for e in terms of the original height h and the maximum height h' reached after one collision.

A wooden block is cut into two pieces, one with three times the mass of the other. A depression is made in both faces of the cut. so that a firecracker can be placed in it with the block reassembled. The reassembled block is set on a rough-surfaced table, and the fuse is lit. When the firecracker explodes, the two blocks separate and slide apart. What is the ratio of distances each block travels?

A 0-kg object moving in the +.v direction at 5.5 m/s collides head-on with a 10.0-kg object moving in the -.v direction at 4.0 m/s. Find the final velocity of each mass if: (a) the objects stick together; (b) the collision is elastic; (c) the 0-kg object is at rest after the collision: (d) the 10.0-kg object is at rest after the collision; (e) the 0-kg object has a velocity of 4.0 m/s in the -x direction after the collision. Are the results in (c), (</), and (e) "reasonable? Explain.

A radioactive nucleus at rest decays into a second nucleus, an electron, and a neutrino. The electron and neutrino arc emitted at right angles and have momenta of 9.30 x 10-2kg-m/s and 5.40 X 10-23kg- m/s. respectively. What are the magnitude and direction of the momentum of the second (recoiling) nucleus?

An eagle (wA = 4.3 kg) moving with speed vA = 7.8 m/s is on a collision course with a second eagle (mB = 5.6 kg) moving at i>u = 10.2 m/s in a direction perpendicular to the first. After they collide, they hold onto one another. In what direction, and with what speed, are they moving after the collision?

Billiard ball A of mass mA = 0.400kg moving with speed vA = 80 m/s strikes ball B. initially at rest, of mass wB = 0.500 kg. As a result of the collision, ball A is deflected off at an angle of 30.0 with a speed v'A = 10 m/s. (A) Taking the x axis to be the original direction of motion of ball A. wrrite down the equations expressing the conservation of momentum for the components in the x and y directions separately. (b) Solve these equations for the speed vjj and angle 0B of ball B. Do not assume the collision is elastic.

After a completely inelastic collision between two objects of equal mass, each having initial speed v, the two move off together with speed ?;/3. What wras the angle between their initial directions?

Two billiard balls of equal mass move at right angles and meet at the origin of an xy coordinate system. Ball A is moving upward along the y axis at 2.0 m/s, and ball B is moving to the right along the x axis with speed 3.7 m/s. After the collision, assumed elastic, ball B is moving along the positive y axis (Fig. 7-37). What is the final direction of ball A and what are their twro speeds?

A neon atom (m = 20.0 u) makes a perfectly elastic collision with another atom at rest. After the impact, the neon atom travels away at a 55.6 angle from its original direction and the unknown atom travels awray at a -50.0 angle. What is the mass (in u) of the unknown atom? f Hint: You can use the law' of sines. |

Find the center of mass of the three-mass system shown in Fig. 7- Specify relative to the left-hand 00- kg mass.

The distance between a carbon atom (mc = 12 u) and an oxygen atom (wiq = 16 u) in the CO molecule is 13 X 10-I0m. How far from the carbon atom is the center of mass of the molecule?

The cm of an empty 1050-kg car is 2.50 m behind the front of the car. How far from the front of the car will the CM be when two people sit in the front seat 2.80 m from the front of the car, and three people sit in the back seat 3.90 m from the front? Assume that each person has a mass of 70.0 kg.

A square uniform raft, 18 m by 18 m, of mass 6800 kg, is used as a ferryboat. If three cars, each of mass 1200 kg. occupy its NF SF., and SVV corners, determine the CM of the loaded ferryboat.

Three cubes, of sides 1$, 2V0. and 3/<>, are placed next to one another (in contact) with their centers along a straight line and the / = 2/0 cube in the center (Fig. 7-39). What is the position, along this line, of the CM of this system? Assume the cubes are made of the same uniform material.

A (lightweight) pallet has a load of identical cases of tomato paste (see Fig. 7-40), each of which is a cube of length /. Find the center of gravity in the horizontal plane, so that the crane operator can pick up the load without tipping it.

A uniform circular plate of radius 2R has a circular hole of radius R cut out of it. The center C' of the smaller circle is a distance 0.80/? from the center C of the larger circle. Fig. 7-4 What is the position of the center of mass of the plate? (Hint:Try subtraction.]

Assume that your proportions are the same as those in Table 7- and calculate the mass of one of your legs.

Determine the cm of an outstretched arm using Table 7-1

Use Table 7-1 to calculate the position of the CM of an arm bent at a right angle. Assume that the person is 155 cm tall.

When a high jumper is in a position such that his arms and legs are hanging vertically, and his trunk and head are horizontal, calculate how far below the torsos median line the cm will be. Will this CM be outside the bodv? Use Table 7-

The masses of the Farth and Moon are 5.98 x 1024 kg and 7.35 X 1022kg. respectively, and their centers are separated by 3.84 x 10sm. (a) Where is the CM of this system located? (b) What can you say about the motion of the F.arth-Moon system about the Sun, and of the Earth and Moon separately about the Sun?

A 55-kg woman and an 80-kg man stand 10.0 m apart on frictionless ice. (rr) How far from the woman is their CM? (b) If each holds one end of a rope, and the man pulls on the rope so that he moves 2.5 m, how far from the woman will he be now? (c) How far will the man have moved when he collides with the woman?

A mallet consists of a uniform cylindrical head of mass 2.00 kg and a diameter 0.0800 m mounted on a uniform cylindrical handle of mass 0.500 kg and length 0.240 m. as shown in Fig. 7-42. If this mallet is tossed, spinning, into the air. how far above the bottom of the handle is the point that will follow a parabolic trajectory?

Suppose that in Example 7-14 (Fig. 7-29), mu = 3mj. Where then would mn land? (/>) What if W| = 3mu?

A helium balloon and its gondola, of mass A/, are in the air and stationary with respect to the ground. A passenger, of mass m. then climbs out and slides down a rope with speed u, measured with respect to the balloon. With what speed and direction (relative to Earth) does the balloon then move? What happens if the passenger stops?

A 0.145-kg baseball pitched horizontally at 35.0 m/s strikes a bat and is popped straight up to a height of 55.6 m. If the contact time is 4 ms. calculate the average force on the ball during the contact.

A rocket of mass m traveling with speed ?>0 along the x axis suddenly shoots out fuel, equal to one- third of its mass, parallel to the v axis (perpendicular to the rocket as seen from the ground) with speed 2vq. Give the components of the final velocity of the rocket.

A novice pool player is faced with the corner pocket shot shown in Fig. 7-43. Relative dimensions are also shown. Should the player be worried about this being a scratch shot. in which the cue ball will also fall into a pocket? Give details.

A 140-kg astronaut (including space suit) acquires a speed of 2.50 m/s by pushing off with his legs from an 1800-kg space capsule, (a) What is the change in speed of the space capsule? (b) If the push lasts 0.40 s. what is the average force exerted on the astronaut by the space capsule? As the reference frame, use the position of the space capsule before the push.

Two astronauts, one of mass 60 kg and the other 80 kg. are initially at rest in outer space.They then push each other apart. How far apart are they when the lighter astronaut has moved 12 m?

A ball of mass m makes a head-on elastic collision with a second ball (at rest) and rebounds in the opposite direction with a speed equal to one-fourth its original speed. What is the mass of the second ball?

You have been hired as an expert witness in a court case involving an automobile accident. The accident involved car A of mass 1900 kg which crashed into stationary car B of mass 1100 kg. The driver of car A applied his brakes 15 m before he crashed into car B. After the collision, car A slid 18 m while car B slid 30 m. 'ITie coefficient of kinetic friction between the locked wheels and the road was measured to be 0.60. Show that the driver of car A was exceeding the 55-mph (90 km/h) speed limit before applying the brakes.

A golf ball rolls off the top of a flight of concrete steps of total vertical height 4.00 m.The ball hits four times on the way down, each time striking the horizontal part of a different step 0 m lower. If all collisions are perfectly elastic, what is the bounce height on the fourth bounce when the ball reaches the bottom of the stairs?

A bullet is fired vertically into a 40-kg block of wood at rest directly above it. If the bullet has a mass of 29.0 g and a speed of 510 m/s. how high will the block rise after the bullet becomes embedded in it?

A 25-g bullet strikes and becomes embedded in a 35-kg block of wood placed on a horizontal surface just in front of the gun. If the coefficient of kinetic friction between the block and the surface is 0.25. and the impact drives the block a distance of 9.5 m before it comes to rest, what was the muzzle speed of the bullet?

Two people, one of mass 75 kg and the other of mass 60 kg. sit in a rowboat of mass 80 kg. With the boat initially at rest, the two people, who have been sitting at opposite ends of the boat 3.2 m apart from each other, now exchange seats. How far and in what direction will the boat move?

A meteor whose mass was about 0 x 10s kg struck the Earth (mt = 6.0 X 1024 kg) with a speed of about 15 km/s and came to rest in the F.arth. (a) What was the Earth's recoil speed? (b) What fraction of the meteor's kinetic energy was transformed to kinetic energy of the Earth? (c) By how much did the Earth's kinetic energy change as a result of this collision?

An object at rest is suddenly broken apart into two fragments by an explosion. One fragment acquires twice the kinetic energy of the other. What is the ratio of their masses?

The force on a bullet is given by the formula F = 580 - (8 X lO71)/ over the time interval t = 0 to / = 3.0 x 10-3s. In this formula, t is in seconds and F is in newtons, (a) Plot a graph of F vs. / for t = 0 to / = 3.0 ms. (b) Estimate, using graphical methods, the impulse given the bullet, (c) If the bullet achieves a speed of 220 m/s as a result of this impulse, given to it in the barrel of a gun. what must its mass be?

Two balls, of masses mA = 40 g and mB = 60 g, are suspended as shown in Fig. 7-44. The lighter ball is pulled away to a 60 angle with the vertical and released. (a) What is the velocity of the lighter ball before impact? (b) What is the velocity of each ball after the elastic collision? (c) What will be the maximum height of each ball after the elastic collision?

An atomic nucleus at rest decays radioactively into an alpha particle and a smaller nucleus. What will be the speed of this recoiling nucleus if the speed of the alpha particle is 3.8 X l(r m/s? Assume the recoiling nucleus has a mass 57 times greater than that of the alpha particle.

A 0.25-kg skeet (clay target) is fired ai an angle of 30" to the horizon with a speed of 25 m/s (Fig. 7- 45). When it reaches the maximum height, it is hit from below by a 15-g pellet traveling vertically upward at a speed of 200m/s. The pellet is embedded in the skeet. (a) How much higher did the skeet go up? (b) How much extra distance. A.v, does the skeet travel because of the collision?

A block of mass m = 2.20 kg slides down a 30.0 incline which is 3.60 m high. At the bottom, it strikes a block of mass M = 7.00 kg which is at rest on a horizontal surface. Fig. 7-46. (Assume a smooth transition at the bottom of the incline.) If the collision is elastic, and friction can be ignored, determine (a) the speeds of the two blocks after the collision, and (b) how far back up the incline the smaller mass will go.

In Problem 79 (Fig. 7-46), what is the upper limit on mass m if it is to rebound from M. slide up the incline, stop, slide down the incline, and collide with A/ again?

The gravitational slingshot effect. Figure 7-47 shows the planet Saturn moving in the negative x direction at its orbital speed (with respect to the Sun) of 9.6km/s. The mass of Saturn is 5.69 x 1026kg. A spacecraft with mass 825 kg approaches Saturn. When far from Saturn, it moves in the +.v direction at 10.4 km/s. The gravitational attraction of Saturn (a conservative force) acting on the spacecraft causes it to swing around the planet (orbit shown as dashed line) and head off in the opposite direction. F.stimate the final speed of the spacecraft after it is far enough away to be considered free of Saturns gravitational pull.

A squash ball hits a wall at a 45° angle as shown in Fig. 7–30. What is the direction (a) of the change in momentum of the ball, (b) of the force on the wall?

Problem 14P A rocket of total mass 3180 kg is traveling in outer space with a velocity of 115 m/s. To alter its course by 35.0°, its rockets can be fired briefly in a direction perpendicular to its original motion. If the rocket gases are expelled at a speed of 1750 m/s, how much mass must be expelled?

Problem 14Q A Superball is dropped from a height h onto a hard steel plate (fixed to the Earth), from which it rebounds at very nearly its original speed. (a) Is the momentum of the ball conserved during any part of this process? (b) If we consider the ball and Earth as our system, during what parts of the process is momentum conserved? (c) Answer part (b) for a piece of putty that falls and sticks to the steel plate.

Problem 15P A golf ball of mass 0.045 kg is hit off the tee at a speed of 45 m/s. The golf club was in contact with the ball for 3.5 × 10–3 s. Find (a) the impulse imparted to the golf ball, and (b) the average force exerted on the ball by the golf club.

Problem 15Q Why do you tend to lean backward when carrying a heavy load in your arms?

(II) Suppose that in Example (Fig. 7-29), Where then would land? What if

Problem 61P A helium balloon and its gondola, of mass M, are in the air and stationary with respect to the ground. A passenger, of mass m, then climbs out and slides down a rope with speed v, measured with respect to the balloon. With what speed and direction (relative to Earth) does the balloon then move? What happens if the passenger stops?

Problem 62GP A 0.145-leg baseball pitched horizontally at 35.0 m/s strikes a bat and is popped straight up to a height of 55.6 m. If the contact time is 1.4 ms, calculate the average force on the ball during the contact.

Problem 63GP A rocket of mass m traveling with speed v0 along the x axis suddenly shoots out fuel, equal to one-third of its mass, parallel to the y axis (perpendicular to the rocket as seen from the ground) with speed 2v0. Give the components of the final velocity of the rocket.

A novice pool player is faced with the corner pocket shot shown in Fig. 7–43. Relative dimensions are also shown. Should the player be worried about this being a “scratch shot,” in which the cue ball will also fall into a pocket? Give details.

Problem 16P A 12-kg hammer strikes a nail at a velocity of 8.5 m/s and comes to rest in a time interval of 8.0 ms. (a) What is the impulse given to the nail? (b) What is the average force acting on the nail?

Problem 16Q Why is the CM of a 1-m length of pipe at its midpoint, whereas this is not true for your arm or leg?

Problem 65GP A 140-kg astronaut (including space suit) acquires a speed of 2.50 m/s by pushing off with his legs from an 1800-kg space capsule. (a) What is the change in speed of the space capsule? (b) If the push lasts 0.40 s, what is the average force exerted on the astronaut by the space capsule? As the reference frame, use the position of the space capsule before the push.

Problem 66GP Two astronauts, one of mass 60 kg and the other 80 kg, are initially at rest in outer space. They then push each other apart. How far apart are they when the lighter astronaut has moved 12 m?

Problem 10Q A light object and a heavy object have the same kinetic energy. Which has the greater momentum? Explain.

Problem 11P An atomic nucleus initially moving at 420 m/s emits an alpha particle in the direction of its velocity, and the remaining nucleus slows to 350 m/s. If the alpha particle has a mass of 4.0 u and the original nucleus has a mass of 222 u, what speed does the alpha particle have when it is emitted?

Problem 11Q Describe a collision in which all kinetic energy is lost.

Problem 12P A 23-g bullet traveling 230 m/s penetrates a 2.0-kg block of wood and emerges cleanly at 170 m/s. If the block is stationary on a frictionless surface when hit, how fast does it move after the bullet emerges?

Problem 12Q At a hydroelectric power plant, water is directed at high speed against turbine blades on an axle that turns an electric generator. For maximum power generation, should the turbine blades be designed so that the water is brought to a dead stop, or so that the water rebounds?

Problem 13P A 975-kg two-stage rocket is traveling at a speed of 5.80 × 103 m/s with respect to Earth when a pre-designed explosion separates the rocket into two sections of equal mass that then move at a speed of 2.20 × 103 m/s relative to each other along the original line of motion. (a) What are the speed and direction of each section (relative to Earth) after the explosion? (b) How much energy was supplied by the explosion? [Hint: What is the change in ke as a result of the explosion?]

(I) Determine the CM of an outstretched arm using Table 7-1

(II) Use Table 7-1 to calculate the position of the CM of an arm bent at a right angle. Assume that the person is 155 cm tall.

(II) A mallet consists of a uniform cylindrical head of mass 2.00 kg and a diameter 0.0800 m mounted on a uniform cylindrical handle of mass 0.500 kg and length 0.240 m, as shown in Fig. 7-42. If this mallet is tossed, spinning, into the air, how far above the bottom of the handle is the point that will follow a parabolic trajectory? Figure 7-42

(II) A tennis ball of mass m = 0.060 kg and speed v = 25 m/s strikes a wall at a \(45^{\circ}\) angle and rebounds with the same speed at \(45^{\circ}\) (Fig. 7-32). What is the impulse (magnitude and direction) given to the ball? Figure 7-32 Equation Transcription: Text Transcription: 45 degrees

Problem 17Q Show on a diagram how your CM shifts when you move from a lying position to a sitting position.

Problem 18P You are the design engineer in charge of the crashworthiness of new automobile models. Cars are tested by smashing them into fixed, massive barriers at 50 km/h (30mph). A new model of mass 1500 kg takes 0.15/s from the time of impact until it is brought to rest. (a) Calculate the average force exerted on the car by the barrier. (b) Calculate the average deceleration of the car.

Problem 18Q If only an external force can change the momentum of the center of mass of an object, how can the internal force of the engine accelerate a car?

Problem 19P A 95-kg fullback is running at 4.0 m/s to the east and is stopped in 0.75 s by a head-on tackle by a tackier running due west. Calculate (a) the original momentum of the fullback, (b) the impulse exerted on the fullback, (c) the impulse exerted on the tackier, and (d) the average force exerted on the tackier.

Problem 68GP You have been hired as an expert witness in a court case involving an automobile accident. The accident involved car A of mass 1900 kg which crashed into stationary car B of mass 1100 kg. The driver of car A applied his brakes 15 m before he crashed into car B. After the collision, car A slid 18 m while car B slid 30 m. The coefficient of kinetic friction between the locked wheels and the road was measured to be 0.60. Show that the driver of car A was exceeding the 55-mph (90 km/h) speed limit before applying the brakes.

Problem 70GP A bullet is fired vertically into a 1.40-kg block of wood at rest directly above it. If the bullet has a mass of 29.0 g and a speed of 510 m/s, how high will the block rise after the bullet becomes embedded in it?

Problem 69GP A golf ball rolls off the top of a flight of concrete steps of total vertical height 4.00 m. The ball hits four times on the way down, each time striking the horizontal part of a different step 1.00 m lower. If all collisions are perfectly elastic, what is the bounce height on the fourth bounce when the ball reaches the bottom of the stairs?

Problem 71GP A 25-g bullet strikes and becomes embedded in a 1.35-kg block of wood placed on a horizontal surface just in front of the gun. If the coefficient of kinetic friction between the block and the surface is 0.25, and the impact drives the block a distance of 9.5 m before it comes to rest, what was the muzzle speed of the bullet?

Problem 67GP A ball of mass m makes a head-on elastic collision with a second ball (at rest) and rebounds in the opposite direction with a speed equal to one-fourth its original speed. What is the mass of the second ball?

Problem 19Q A rocket following a parabolic path through the air suddenly explodes into many pieces. What can you say about the motion of this system of pieces?

(II) Suppose the force acting on a tennis ball (mass ) points in the direction and is given by the graph of Fig. as a function of time. Use graphical methods to estimate the total impulse given the ball, and the velocity of the ball after being struck, assuming the ball is being served so it is nearly at rest initially.

Problem 21P From what maximum height can a 75-kg person jump without breaking the lower leg bone of either leg? Ignore air resistance and assume the cm of the person moves a distance of 0.60 m from the standing to the seated position (that is, in breaking the fall). Assume the breaking strength (force per unit area) of bone is 170 × 106 N/m2, and its smallest cross-sectional area is 2.5 × 10-4 m2. [Hint: Do not try this experimentally.]

Problem 23P A 0.450-kg ice puck, moving east with a speed of 3.00 m/s, has a head-on collision with a 0.900-kg puck initially at rest. Assuming a perfectly elastic collision, what will be the speed and direction of each object after the collision?

Problem 22P A ball of mass 0.440 kg moving east (+ x direction) with a speed of 3.30 m/s collides head-on with a 0.220-kg ball at rest. If the collision is perfectly elastic, what will be the speed and direction of each ball after the collision?

Problem 72GP Two people, one of mass 75 kg and the other of mass 60kg, sit in a rowboat of mass 80kg. With the boat initially at rest, the two people, who have been sitting at opposite ends of the boat 3.2 m apart from each other, now exchange seats. How far and in what direction will the boat move?

Problem 73GP A meteor whose mass was about 1.0 × 108 kg struck the Earth (mE = 6.0 × 1024 kg) with a speed of about 15 km/s and came to rest in the Earth. (a) What was the Earth’s recoil speed? (b) What fraction of the meteor’s kinetic energy was transformed to kinetic energy of the Earth? (c) By how much did the Earth’s kinetic energy change as a result of this collision?

Problem 74GP (II) An object at rest is suddenly broken apart into two fragments by an explosion. One fragment acquires twice the kinetic energy of the other. What is the ratio of their masses?

Problem 75GP The force on a bullet is given by the formula F = 580 – (l.8 × 105)t over the time interval t = 0 to t= 3.0 × 10–3s. in this formula, t is in seconds and F is in newtons, (a) Plot a graph of F vs. t for t= 0 to t = 3.0 ms.(b) Estimate, using graphical methods, the impulse given the bullet, (c) If the bullet achieves a speed of 220 m/s as a result of this impulse, given to it in the barrel of a gun, what must its mass be?

Two balls, of masses \(m_{\mathrm{A}}=40\mathrm{\ g}\) and \(m_{\mathrm{B}}=60\mathrm{\ g}\), are suspended as shown in Fig. 7-44. The lighter ball is pulled away to a \(60^{\circ}\) angle with the vertical and released. (a) What is the velocity of the lighter ball before impact? (b) What is the velocity of each ball after the elastic collision? (c) What will be the maximum height of each ball after the elastic collision? Figure 7-44 Equation Transcription: Text Transcription: m_A = 40 g m_B = 60 g 60 degrees

Problem 24P Two billiard balls of equal mass undergo a perfectly elastic head-on collision. If one ball’s initial speed was 2.00 /s, and the other’s was 3.00 m/s in the opposite direction, what will be their speeds after the collision?

Problem 25P A 0.060-kg tennis ball, moving with a speed of 2.50 m/s, collides head-on with a 0.090-kg ball initially moving away from it at a speed of 1.15 m/s. Assuming a perfectly elastic collision, what are the speed and direction of each ball after the collision?

Problem 26P A softball of mass 0.220 kg that is moving with a speed of 8.5 m/s collides head-on and elastically with another ball initially at rest. Afterward the incoming soft-ball bounces backward with a speed of 3.7 m/s. Calculate (a)the velocity of the target ball after the collision, and (b) the mass of the target ball.

Two bumper cars in an amusement park ride collide elastically as one approaches the other directly from the rear (Fig. 7–34). Car A has a mass of 450 kg and car B 550 kg, owing to differences in passenger mass. If car A approaches at 4.50 m/s and car B is moving at 3.70 m/s, calculate (a) their velocities after the collision, and (b) the change in momentum of each.

Problem 28P A 0.280-kg croquet ball makes an elastic head-on collision with a second ball initially at rest. The second ball moves off with half the original speed of the first ball. (a) What is the mass of the second ball? (b) What fraction of the original kinetic energy (?KE/KE) gets transferred to the second ball?

An atomic nucleus at rest decays radioactively into an alpha particle and a smaller nucleus. What will be the speed of this recoiling nucleus if the speed of the alpha particle is 3.8 × 105m/s? Assume the recoiling nucleus has a mass 57 times greater than that of the alpha particle.

A 0.25-kg skeet (clay target) is fired at an angle of \(30^{\circ}\) to the horizon with a speed of 25 m/s (Fig. 7-45). When it reaches the maximum height, it is hit from below by a 15-g pellet traveling vertically upward at a speed of 200 m/s. The pellet is embedded in the skeet. (a) How much higher did the skeet go up? (b) How much extra distance, \(\Delta x\), does the skeet travel because of the collision? Figure 7-45 Equation Transcription: Text Transcription: 60 degrees Delta x

A block of mass m = 2.20 kg slides down a \(30.0^{\circ}\) incline which is 3.60 m high. At the bottom, it strikes a block of mass M = 7.00 kg which is at rest on a horizontal surface, Fig. 7-46. (Assume a smooth transition at the bottom of the incline.) If the collision is elastic, and friction can be ignored, determine (a) the speeds of the two blocks after the collision, and (b) how far back up the incline the smaller mass will go. Figure 7-46 Equation Transcription: Text Transcription: 30.0 degrees

In Problem 79 (Fig. 7–46.), what is the upper limit on mass m if it is to rebound from M, slide up the incline, stop, slide down the incline, and collide with M again?

The gravitational slingshot effect. Figure shows the planet Saturn moving in the negative direction at its orbital speed (with respect to the Sun) of . The mass of Saturn is A spacecraft with mass approaches Saturn. When far from Saturn, it moves in the direction at . The gravitational attraction of Saturn (a conservative force) acting on the spacecraft causes it to swing around the planet (orbit shown as dashed line) and head off in the opposite direction. Estimate the final speed of the spacecraft after it is far enough away to be considered free of Saturn's gravitational pull. Figure 7-47 Problem 81.

Problem 57P (II) The masses of the Earth and Moon are 5.98 X 1024 kg And 7.35 X 1022 kg, respectively, and their centers are separated by 3.84 X 108 m. (a) Where is the CMof the Earth–Moon system located? (b) What can you say about the motion of the Earth–Moon system about the Sun, and of the Earth and Moon separately about the Sun?

In a physics lab, a cube slides down a frictionless incline as shown in Fig. 7–35 and elastically strikes another cube at the bottom that is only one-half its mass. If the incline is 30 cm high and the table is 90 cm off the floor, where does each cube land? [Hint: Both leave the incline moving horizontally.]

(III) Take the general case of an object of mass and velocity elastically striking a stationary object of mass head-on. Show that the final velocities and are given by (b) What happens in the extreme case when is much smaller than ? Cite a common example of this. (c) What happens in the extreme case when is much larger than ? Cite a common example of this. What happens in the case when ? Cite a common example.

Problem 31P (I) In a ballistic pendulum experiment, projectile 1 results in a maximum height h of the pendulum equal to 2.6 cm. A second projectile (of the same mass) causes the pendulum to swing twice as high, h2 =5.2 cm. The second projectile was how many times faster than the first?

Problem 32P A 28-g rifle bullet traveling 230 m/s buries itself in a 3.6-kg pendulum hanging on a 2.8-m-long string, which makes the pendulum swing upward in an arc. Determine the vertical and horizontal components of the pendulum’s displacement.

(II) (a) Derive a formula for the fraction of kinetic energy lost, , for the ballistic pendulum collision of Example (b) Evaluate for and .

Problem 1P (I) What is the magnitude of the momentum of a 28-g sparrow flying with a speed of 8.4 m/s?

Problem 1Q We claim that momentum is conserved. Yet most moving objects eventually slow down and stop. Explain.

Problem 2P (I) A constant friction force of 25 N acts on a 65-kg skier for 20 s. What is the skier’s change in velocity?

Problem 2Q When a person jumps from a tree to the ground, what happens to the momentum of the person upon striking the ground?

Problem 34P An internal explosion breaks an object, initially at rest, into two pieces, one of which has 1.5 times the mass of the other. If 7500 J were released in the explosion, how much kinetic energy did each piece acquire?

Problem 35P A 920-kg sports car collides into the rear end of a 2300-kg SUV stopped at a red light. The bumpers lock, the brakes are locked, and the two cars skid forward 2.8 m before stopping. The police officer, knowing that the coefficient of kinetic friction between tires and road is 0.80, calculates the speed of the sports car at impact. What was that speed?

Problem 36P A ball is dropped from a height of 1.50 m and rebounds to a height of 1.20 m. Approximately how many rebounds will the ball make before losing 90% of its energy?

(II) A measure of inelasticity in a head-on collision of two objects is the coefficient of restitution, \(e\), defined as \(e=\frac{v_{\mathrm{A}}^{\prime}-v_{\mathrm{B}}^{\prime}}{v_{\mathrm{B}}-v_{\mathrm{A}}}\), where \(v_{\mathrm{A}}^{\prime}-v_{\mathrm{B}}^{\prime}\) is the relative velocity of the two objects after the collision and \(v_{\mathrm{B}}-v_{\mathrm{A}}\) is their relative velocity before it. (a) Show the e = 1 for a perfectly elastic collision, and e = 0 for a completely inelastic collision. (b) A simple method for measuring the coefficient of restitution for an object colliding with a very hard surface like steel is to drop the object onto a heavy steel plate, as shown in Fig. 7-36. Determine the formula for \(e\) in terms of the original height \(h\) and the maximum height \(h^{\prime}\) reached after one collision. Figure 7-36 Measurement of the coefficient of restitution Equation Transcription: Text Transcription: e e = v'_A - v'_B / v_B - v_A v'_A - v'_B v_B - v_A h h'

Problem 38P (II) A wooden block is cut into two pieces, one with three times the mass of the other. A depression is made in both faces of the cut, so that a firecracker can be placed in it with the block reassembled. The reassembled block is set on a rough-surfaced table, and the fuse is lit .When the firecracker explodes inside, the two blocks separate and slide apart. What is the ratio of distances each block travels?

Problem 58P A 55-kg woman and an 80-kg man stand 10.0 m apart on frictionless ice. (a) How far from the woman is their CM? (b) If each holds one end of a rope, and the man pulls on the rope so that he moves 2.5 m, how far from the woman will he be now? (c) How far will the man have moved when he collides with the woman?

Problem 3P A 0.145-kg baseball pitched at 39.0 m/s is hit on a horizontal line drive straight back toward the pitcher at 52.0 m/s. If the contact time between bat and ball is 3.00 × 10–3 s, calculate the average force between the ball and bat during contact.

Problem 3Q When you release an inflated but untied balloon, why does it fly across the room?

(II) A child in a boat throws a 6.40-kg package out horizontally with a speed of 10.0 m/s, Fig. 7–31. Calculate the velocity of the boat immediately after, assuming it was initially at rest. The mass of the child is 26.0 kg, and that of the boat is 45.0 kg. Ignore water resistance.

Problem 4Q It is said that in ancient times a rich man with a bag of gold coins was stranded on the surface of a frozen lake. Because the ice was frictionless, he could not push himself to shore and froze to death. What could he have done to save himself had he not been so miserly?

Problem 5P Calculate the force exerted on a rocket, given that the propelling gases are expelled at a rate of 1500 kg/s with a speed of 4.0 × 104 m/s (at the moment of takeoff).

Problem 40P A radioactive nucleus at rest decays into a second nucleus, an electron, and a neutrino. The electron and neutrino are emitted at right angles and have momenta of 9.30 × 10–23 kg·m/s and 5.40 × 10–23 kg·m/s, respectively. What are the magnitude and direction of the momentum of the second (recoiling) nucleus?

Problem 41P An eagle (mA = 4.3 kg) moving with speed vA = 7.8 m/s is on a collision course with a second eagle (mB = 5.6 kg) moving at vB = 10.2 m/s in a direction perpendicular to the first. After they collide, they hold onto one another. In what direction, and with what speed, are they moving after the collision?

Problem 42P Billiard ball A of mass mA = 0.400 kg moving with speed vA = 1.80 m/s strikes ball B, initially at rest, of mass mB = 0.500 kg. As a result of the collision, ball A is deflected off at an angle of 30.0° with a speed v?A = 1.10 m/s. (a) Taking the x axis to be the original direction of motion of ball A, write down the equations expressing the conservation of momentum for the components in the x and y directions separately. (b) Solve these equations for the speed v?B and angle ??B of ball B. Do not assume the collision is elastic.

Problem 43P After a completely inelastic collision between two objects of equal mass, each having initial speed v, the two move off together with speed v/3. What was the angle between their initial directions?

Problem 39P A 15.0-kg object moving in the + x direction at 5.5 m/s collides head-on with a 10.0-kg object moving in the –x direction at 4.0 m/s. Find the final velocity of each mass if: (a) the objects stick together; (b) the collision is elastic; (c) the 15.0-kg object is at rest after the collision; (d) the 10.0-kg object is at rest after the collision; (e) the 15.0-kg object has a velocity of 4.0 m/s in the –xdirection after the collision. Are the results in (c), (d) and (e) “reasonable”? Explain.

(II) When a high jumper is in a position such that his arms and lower legs are hanging vertically, and his thighs, trunk, and head are horizontal, calculate how far below the torso’s median line the CM will be. Will this CM be outside the body? Use Table 7–1

Problem 5Q How can a rocket change direction when it is far out in space and essentially in a vacuum?

Problem 6P A 95-kg halfback moving at 4.1 m/s on an apparent breakaway for a touchdown is tackled from behind. When he was tackled by an 85-kg cornerback running at 5.5 m/s in the same direction, what was their mutual speed immediately after the tackle?

According to Eq. 7–4, the longer the impact time of an impulse, the smaller the force can be for the same momentum change, and hence the smaller the deformation of the object on which the force acts. On this basis, explain the value of air bags, which are intended to inflate during an automobile collision and reduce the possibility of fracture or death.

Problem 7P A 12,600-kg railroad car travels alone on a level frictionless track with a constant speed of 18.0 m/s. A 5350-kg load, initially at rest, is dropped onto the car. What will be the car’s new speed?

Problem 7Q 15. Cars used to be built as rigid as possible to withstand collisions. Today, though, cars are designed to have “crumple zones” that collapse upon impact. What is the advantage of this new design?

(III) Two billiard balls of equal mass move at right angles and meet at the origin of an \(xy\) coordinate system. Ball \(A\) is moving upward along the \(y\) axis at 2.0 m/s, and ball \(B\) is moving to the right along the \(x\) axis with speed 3.7 m/s. After the collision, assumed elastic, ball \(B\) is moving along the positive \(y\) axis (Fig. 7-37). What is the final direction of ball \(A\) and what are their two speeds? Figure 7-37 Equation Transcription: Text Transcription: xy A y B x

(I) Find the center of mass of the three-mass system shown in Fig. 7-38. Specify relative to the left-hand 1.00-kg mass. Figure 7-38

Problem 47P (I) The distance between a carbon atom (mC =12 u) and an oxygen atom (mO =16 u) in the CO molecule is 1.13 X 10-10 m. How far from the carbon atom is the center of mass of the molecule?

Problem 48P (I) The CM of an empty 1050-kg car is 2.50 m behind the front of the car. How far from the front of the car will the CM be when two people sit in the front seat 2.80 m from the front of the car, and three people sit in the back seat 3.90 m from the front? Assume that each person has a mass of 70.0 kg.

Problem 45P (III) A neon atom (m =20.0 u) makes a perfectly elastic collision with another atom at rest. After the impact, the neon atom travels away at a 55.6° angle from its original direction and the unknown atom travels away at a -50.0° angle. What is the mass (in u) of the unknown atom? [Hint: You could use the law of sines.]

Problem 8P A 9300-kg boxcar traveling at 15.0 m/s strikes a second boxcar at rest. The two stick together and move off with a speed of 6.0 m/s. What is the mass of the second car?

Problem 8Q Why can a batter hit a pitched baseball farther than a ball he himself has tossed up in the air?

Problem 9P During a Chicago storm, winds can whip horizontally at speeds of 100 km/h. If the air strikes a person at the rate of 40 kg/s per square meter and is brought to rest, estimate the force of the wind on a person. Assume the person is 1.50 m high and 0.50 m wide. Compare to the typical maximum force of friction (? ? 1.0) between the person and the ground, if the person has a mass of 70 kg.

Problem 9Q Is it possible for an object to receive a larger impulse from a small force than from a large force? Explain.

Problem 10P A 3800-kg open railroad car coasts along with a constant speed of 8.60 m/s on a level track. Snow begins to fall vertically and fills the car at a rate of 3.50kg/min. Ignoring friction with the tracks, what is the speed of the car after 90.0 min?

Problem 49P A square uniform raft, 18 m by 18 m, of mass 6800 kg, is used as a ferryboat. If three cars, each of mass 1200 kg, occupy its NE, SE, and SW corners, determine the CM of the loaded ferryboat.

Three cubes, of sides , , and , are placed next to one another (in contact) with their centers along a straight line and the cube in the center (Fig. 7–39). What is the position, along this line, of the CM of this system? Assume the cubes are made of the same uniform material. Figure 7–39

(II) A (lightweight) pallet has a load of ten identical cases of tomato paste (see Fig. 7–39), each of which is a cube of length l Find the center of gravity in the horizontal plane, so that the crane operator can pick up the load without tipping it. Figure 7-40 Problem 51

(III) A uniform circular plate of radius 2R has a circular hole of radius R cut out of it. The center C of the smaller circle is a distance 0.80R from the center C of the larger circle, Fig. 7–41. What is the position of the center of mass of the plate? [Hint: Try subtraction.] Figure 7-41 Problem 52

(I) Assume that your proportions are the same as those in Table 7–1, and calculate the mass of one of your legs.