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Chapter 7

Linear Momentum

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Problem 1

(I) What is the magnitude of the momentum of a 28-g sparrow flying with a speed of 8.4 $\mathrm{m} / \mathrm{s} ?$

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Problem 2

(I) A constant friction force of 25 $\mathrm{N}$ acts on a $65-\mathrm{kg}$ skier for 20 $\mathrm{s}$ . What is the skier's change in velocity?

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Problem 3

(II) A 0.145 -kg baseball pitched at 39.0 $\mathrm{m} / \mathrm{s}$ is hit on a horizontal line drive straight back toward the pitcher at 52.0 $\mathrm{m} / \mathrm{s}$ . If the contact time between bat and ball is $3.00 \times 10^{-3} \mathrm{s}$ , calculate the average force between the ball and bat during contact.

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Problem 4

(II) A child in a boat throws a 6.40 -kg package out horizontally with a speed of 10.0 $\mathrm{m} / \mathrm{s}$ , 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 $\mathrm{kg}$ , and that of the boat is 45.0 $\mathrm{kg}$ . Ignore water resistance.

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Problem 5

(II) Calculate the force exerted on a rocket, given that the propelling gases are expelled at a rate of 1500 $\mathrm{kg} / \mathrm{s}$ with a speed of $4.0 \times 10^{4} \mathrm{m} / \mathrm{s}$ (at the moment of takeoff).

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Problem 6

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

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Problem 7

(II) A $12,600$ -kg railroad car travels alone on a level frictionless track with a constant speed of 18.0 $\mathrm{m} / \mathrm{s}$ . A $5350-\mathrm{kg}$ load, initially at rest, is dropped onto the car. What will be the car's new speed?

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Problem 8

(II) A $9300-\mathrm{kg}$ boxcar traveling at 15.0 $\mathrm{m} / \mathrm{s}$ strikes a second boxcar at rest. The two stick together and move off with a speed of 6.0 $\mathrm{m} / \mathrm{s}$ . What is the mass of the second car?

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Problem 9

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

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Problem 10

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

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Problem 11

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

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Problem 12

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

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Problem 13

(III) A $975-\mathrm{kg}$ two-stage rocket is traveling at a speed of $5.80 \times 10^{3} \mathrm{m} / \mathrm{s}$ 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 $2.20 \times 10^{3} \mathrm{m} / \mathrm{s}$ 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? ]

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Problem 14

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

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Problem 15

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

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Problem 16

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

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Problem 17

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

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Problem 18

(II) 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 $\mathrm{km} / \mathrm{h}$ (30 mph). A new model of mass 1500 $\mathrm{kg}$ takes 0.15 $\mathrm{s}$ 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.

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Problem 19

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

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Problem 20

(II) Suppose the force acting on a tennis ball (mass 0.060 $\mathrm{kg}$ ) points in the $+x$ direction and is given by the graph of Fig. $7-33$ as as a function of time. Use graphical methods to estimate (a) 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.

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Problem 21

(III) From what maximum height can a $75-\mathrm{kg}$ 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 $\mathrm{m}$ from the standing to the seated position (that is, in breaking the fall). Assume the breaking strength (force per unit area) of bone is $170 \times 10^{6} \mathrm{N} / \mathrm{m}^{2},$ and its smallest cross-sectional area is $2.5 \times 10^{-4} \mathrm{m}^{2} .[\text { Hint: Do not try this experimentally. }]$

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Problem 22

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

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Problem 23

(II) A $0.450-\mathrm{kg}$ ice puck, moving east with a speed of $3.00 \mathrm{m} / \mathrm{s},$ 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?

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Problem 24

(II) Two billiard balls of equal mass undergo a perfectly elastic head-on collision. If one ball's initial speed was 2.00 $\mathrm{m} / \mathrm{s}$ , and the other's was 3.00 $\mathrm{m} / \mathrm{s}$ in the opposite direction, what will be their speeds after the collision?

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Problem 25

(II) A $0.060-\mathrm{kg}$ tennis ball, moving with a speed of 2.50 $\mathrm{m} / \mathrm{s}$ , collides head-on with a $0.090-\mathrm{kg}$ ball initially moving away from it at a speed of 1.15 $\mathrm{m} / \mathrm{s}$ . Assuming a perfectly elastic collision, what are the speed and direction of each ball after the collision?

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Problem 26

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

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Problem 27

(II) 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 $\mathrm{kg}$ and car $\mathrm{B}$ 550 $\mathrm{kg}$ , owing to differences in passenger mass. If car $\mathrm{A}$ approaches at 4.50 $\mathrm{m} / \mathrm{s}$ and car $\mathrm{B}$ is moving at 3.70 $\mathrm{m} / \mathrm{s}$ , calculate $(a)$ their velocities after the collision, and $(b)$ the change in momentum of each.

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Problem 28

(II) A $0.280-\mathrm{kg}$ 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 $(\Delta \mathrm{KE} / \mathrm{KE})$ gets transferred to the second ball?

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Problem 29

(III) 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 $\mathrm{cm}$ high and the table is 90 $\mathrm{cm}$ off the floor, where does each cube land? [Hint: Both leave the incline moving horizontally.

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Problem 30

(III) Take the general case of an object of mass $m_{\mathrm{A}}$ and velocity $v_{\mathrm{A}}$ elastically striking a stationary $\left(v_{\mathrm{B}}=0\right)$ object of mass $m_{\mathrm{B}}$ head-on. ( $a$ ) Show that the final velocities $v_{\mathrm{A}}^{\prime}$ and $v_{\mathrm{B}}^{\prime}$ are given by
$$\begin{aligned} v_{\mathrm{A}}^{\prime} &=\left(\frac{m_{\mathrm{A}}-m_{\mathrm{B}}}{m_{\mathrm{A}}+m_{\mathrm{B}}}\right) v_{\mathrm{A}} \\ v_{\mathrm{B}}^{\prime} &=\left(\frac{2 m_{\mathrm{A}}}{m_{\mathrm{A}}+m_{\mathrm{B}}}\right) v_{\mathrm{A}} \end{aligned}$$
(b) What happens in the extreme case when $m_{\mathrm{A}}$ is much smaller than $m_{\mathrm{B}} ?$ Cite a common example of this.
(c) What happens in the extreme case when $m_{\mathrm{A}}$ is much larger than $m_{\mathrm{B}}$ ? Cite a common example of this. (d) What happens in the case when $m_{\mathrm{A}}=m_{\mathrm{B}} ?$ Cite a common example.

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Problem 31

(I) In a ballistic pendulum experiment, projectile 1 results in a maximum height $h$ of the pendulum equal to 2.6 $\mathrm{cm}$ . A second projectile causes the the pendulum to swing twice as high, $h_{2}=5.2 \mathrm{cm} .$ The second projectile was how many times faster than the first?

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Problem 32

(II) A 28 -g rifle bullet traveling 230 $\mathrm{m} / \mathrm{s}$ 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.

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Problem 33

(II) $(a)$ Derive a formula for the fraction of kinetic energy lost, $\Delta \mathrm{kE} / \mathrm{KE},$ for the ballistic pendulum collision of Example $7-10 .(b)$ Evaluate for $m=14.0 \mathrm{g}$ and $M=380 \mathrm{g}$ .

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Problem 34

(II) 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 $\mathrm{J}$ were released in the explosion, how much kinetic energy did each piece acquire?

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Problem 35

(II) 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 $\mathrm{m}$ 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?

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Problem 36

(II) A ball is dropped from a height of 1.50 $\mathrm{m}$ and rebounds to a height of $1.20 \mathrm{m} .$ Approximately how many rebounds will the ball make before losing 90$\%$ of its energy?

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Problem 37

(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 that $e=1$ for a perfectly elastic collision, and $e=0$ for a completely inelastic collision. ( $b ) \mathrm{A}$ 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 $h^{\prime}$ reached after one collision.

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Problem 38

(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, the two blocks separate and slide apart. What is the ratio of distances each block travels?

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Problem 39

(III) A 15.0 -kg object moving in the $+x$ direction at 5.5 $\mathrm{m} / \mathrm{s}$ collides head-on with a $10.0-\mathrm{kg}$ object moving in the $-x$ direction at 4.0 $\mathrm{m} / \mathrm{s}$ . Find the final velocity of each mass if: $(a)$ the objects stick together; (b) the collision is elastic; $(c)$ the $15.0-\mathrm{kg}$ object is at rest after the collision; (d) the 10.0$\cdot$ kg object is at rest after the collision; $(e)$ the $15.0-\mathrm{kg}$ object has a velocity of 4.0 $\mathrm{m} / \mathrm{s}$ in the $-x$ direction after the collision. Are the results in $(c),(d),$ and $(e)$ "reasonable"? Explain.

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Problem 40

(II) 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 $9.30 \times 10^{-23} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}$ and $5.40 \times 10^{-23} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}$ , respectively. What are the magnitude and direction of the momentum of the second (recoiling) nucleus?

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Problem 41

(II) An eagle $\left(m_{A}=4.3 \mathrm{kg}\right)$ moving with speed $v_{\mathrm{A}}=7.8 \mathrm{m} / \mathrm{s}$ is on a collision course with a second eagle $\left(m_{\mathrm{B}}=5.6 \mathrm{kg}\right)$ moving at $v_{\mathrm{B}}=10.2 \mathrm{m} / \mathrm{s}$ 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?

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Problem 42

(II) Billiard ball A of mass $m_{A}=0.400 \mathrm{kg}$ moving with speed $v_{\mathrm{A}}=1.80 \mathrm{m} / \mathrm{s}$ strikes ball $\mathrm{B}$ , initially at rest, of mass $m_{\mathrm{B}}=0.500 \mathrm{kg}$ . As a result of the collision, ball $\mathrm{A}$ is deflected off at an angle of $30.0^{\circ}$ with a speed $v_{\mathrm{A}}^{\prime}=1.10 \mathrm{m} / \mathrm{s} .(a)$ Taking the $x$ axis to be the original direction of motion of ball $\mathrm{A}$ , 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 $v_{\mathrm{B}}^{\prime}$ and angle $\theta_{\mathrm{B}}^{\prime}$ of ball $\mathrm{B}$ . Do not assume the collision is elastic.

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Problem 43

(III) 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?

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Problem 44

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

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Problem 45

(III) A neon atom $(m=20.0 \mathrm{u})$ makes a perfectly elastic collision with another atom at rest. After the impact, the neon atom travels away at a $55.6^{\circ}$ angle from its original direction and the unknown atom travels away at a $-50.0^{\circ}$ angle. What is the mass (in u) of the unknown atom? [Hint: You can use the law of sines.]

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Problem 46

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

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Problem 47

(I) The distance between a carbon atom $\left(m_{\mathrm{C}}=12 \mathrm{u}\right)$ and an oxygen atom $\left(m_{\mathrm{O}}=16 \mathrm{u}\right)$ in the CO molecule is $1.13 \times 10^{-10} \mathrm{m}$ . How far from the carbon atom is the center of mass of the molecule?

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Problem 48

(I) The distance between a carbon atom $\left(m_{\mathrm{C}}=12 \mathrm{u}\right)$ and an oxygen atom $\left(m_{\mathrm{O}}=16 \mathrm{u}\right)$ in the CO molecule is $1.13 \times 10^{-10} \mathrm{m}$ . How far from the carbon atom is the center of mass of the molecule?

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Problem 49

(II) Three cubes, of sides $l_{0}, 2 l_{0},$ and 3$l_{0}$ , are placed next to one another (in contact) with their centers along a straight line and the $l=2 l_{0}$ 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.

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Problem 50

(II) Three cubes, of sides $l_{0}, 2 l_{0},$ and $3 l_{0},$ are placed next to one another (in contact) with their centers along a straight line and the $l=2 l_{0}$ 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.

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Problem 51

(II) A (lightweight) pallet has a load of identical cases of tomato paste (see Fig. $7-40 )$ , 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.

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Problem 52

(III) A uniform circular plate of radius 2$R$ has a circular hole of radius $R$ cut out of it. The center $C^{\prime}$ of the smaller circle is a distance 0.80$R$ 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.]

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Problem 53

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

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Problem 54

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

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Problem 55

(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 $\mathrm{cm}$ tall.

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Problem 56

(II) 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 torso's median line the CM will be. Will this $\mathrm{CM}$ be outside the body? Use Table $7-1$ .

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Problem 57

(II) The masses of the Earth and Moon are $5.98 \times 10^{24} \mathrm{kg}$ and $7.35 \times 10^{22} \mathrm{kg}$ , respectively, and their centers are separated by $3.84 \times 10^{8} \mathrm{m}$ . (a) Where is the CM of this 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?

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Problem 58

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

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Problem 59

(II) A mallet consists of a uniform cylindrical head of mass 2.00 $\mathrm{kg}$ and a diameter 0.0800 $\mathrm{m}$ mounted on a uniform cylindrical handle of mass 0.500 $\mathrm{kg}$ and length $0.240 \mathrm{m},$ 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?

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Problem 60

(II) $(a)$ Suppose that in Example $7-14$ (Fig. $7-29 )$ , $m_{11}=3 m_{1}$ . Where then would $m_{11}$ land? $(b)$ What if $m_{1}=3 m_{11}$ ?

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Problem 61

(III) 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?

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Problem 62

A 0.145 -kg baseball pitched horizontally at 35.0 $\mathrm{m} / \mathrm{s}$ strikes a bat and is popped straight up to a height of 55.6 $\mathrm{m}$ . If the contact time is 1.4 $\mathrm{ms}$ , calculate the average force on the ball during the contact.

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Problem 63

A rocket of mass $m$ traveling with speed $v_{0}$ 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 $2 v_{0} .$ Give the components of the final velocity of the rocket.

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Problem 64

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.

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Problem 65

A $140-\mathrm{kg}$ astronaut (including space suit) acquires a speed of 2.50 $\mathrm{m} / \mathrm{s}$ by pushing off with his legs from an 1800 -kg space capsule. (a) What is the change in speed of the space capsule? ( $a$ ) If the push lasts 0.40 $\mathrm{s}$ , 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.

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Problem 66

Two astronauts, one of mass 60 $\mathrm{kg}$ and the other 80 $\mathrm{kg}$ , 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 $\mathrm{m} ?$

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Problem 67

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?

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Problem 68

You have been hired as an expert witness in a court case involving an automobile accident. The accident involved car $\mathrm{A}$ of mass 1900 $\mathrm{kg}$ which crashed into stationary car $\mathrm{B}$ of mass 1100 $\mathrm{kg}$ . The driver of car A applied his brakes 15 $\mathrm{m}$ before he crashed into car B. After the collision, You have been hired as an expert witness in a court case involving an automobile accident. The accident involved car $\mathrm{A}$ of mass 1900 $\mathrm{kg}$ which crashed into stationary car $\mathrm{B}$ of mass 1100 $\mathrm{kg}$ . The driver of car A applied his brakes 15 $\mathrm{m}$ before he crashed into car B. After the collision,

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Problem 69

A golf ball rolls off the top of a flight of concrete steps of total vertical height 4.00 $\mathrm{m}$ . The ball hits four times on the way down, each time striking the horizontal part of a different step 1.0 $\mathrm{m}$ 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?

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Problem 70

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

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Problem 71

A $25-\mathrm{g}$ bullet strikes and becomes embedded in a $1.35-\mathrm{kg}$ 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 $\mathrm{m}$ before it comes to rest, what was the muzzle speed of the bullet?

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Problem 72

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

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Problem 73

A meteor whose mass was about $1.0 \times 10^{8} \mathrm{kg}$ struck the Earth $\left(m_{\mathrm{E}}=6.0 \times 10^{24} \mathrm{kg}\right)$ with a speed of about 15 $\mathrm{km} / \mathrm{s}$ 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?

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Problem 74

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?

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Problem 75

The force on a bullet is given by the formula $F=580-\left(1.8 \times 10^{5}\right) t$ over the time interval $t=0$ to $t=3.0 \times 10^{-3} \mathrm{s} .$ In this formula, $t$ is in seconds and $F$ is in newtons. $(a)$ Plot a graph of $F$ vs. $t$ for $t=0$ to $t=3.0 \mathrm{ms}$ . $(b)$ Estimate, using graphical methods, the impulse given the bullet. $(c)$ If the bullet achieves a speed of 220 $\mathrm{m} / \mathrm{s}$ as a result of this impulse, given to it in the barrel of a gun, what must its mass be?

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Problem 76

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?

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Problem 77

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 $3.8 \times 10^{5} \mathrm{m} / \mathrm{s} ?$ Assume the recoiling nucleus has a mass 57 times greater than that of the alpha particle.

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Problem 78

A $0.25-\mathrm{kg}$ skeet (clay target) is fired at an angle of $30^{\circ}$ to the horizon with a speed of 25 $\mathrm{m} / \mathrm{s}$ (Fig. $7-45$ ). When it reaches the maximum height, it is hit from below by a $15-\mathrm{g}$ pellet traveling vertically upward at a speed of 200 $\mathrm{m} / \mathrm{s}$. 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?

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Problem 79

A block of mass $m=2.20 \mathrm{kg}$ slides down a $30.0^{\circ}$ incline which is 3.60 $\mathrm{m}$ high. At the bottom, it strikes a block of mass $M=7.00 \mathrm{kg}$ 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.

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Problem 80

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?

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Problem 81

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 9.6 $\mathrm{km} / \mathrm{s}$ . The mass of Saturn is $5.69 \times 10^{26} \mathrm{kg}$ . A spacecraft with mass 825 $\mathrm{kg}$ approaches Saturn. When far from Saturn, it moves in the $+x$ direction at 10.4 $\mathrm{km} / \mathrm{s}$ . 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. Estimate the final speed of the spacecraft after it is far enough away to be considered free of Saturn's gravitational pull.

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