|| INT A 550 g cart is released from rest on a frictionless, $\text{30}°$ ramp, 120 cm from the bottom of the ramp. It rolls down, bounces off a rubber block at the bottom, and then rolls 80 cm back up the ramp. A high-speed video shows that the cart was in contact with the rubber block for 25 ms. What was the average force exerted on the cart by the block?

|| A tennis player swings her 1000 g racket with a speed of $10\text{m}\text{/}\text{s}$. She hits a 60 g tennis ball that was approaching her at a speed of $20\text{m}\text{/}\text{s}$. The ball rebounds at $40\text{m}\text{/}\text{s}$.

1. How fast is her racket moving immediately after the impact? You can ignore the interaction of the racket with her hand for the brief duration of the collision.

2. If the tennis ball and racket are in contact for 10 ms, what is the average force that the racket exerts on the ball? How does this compare to the gravitational force on the ball?

|| A 60 g tennis ball with an initial speed of $32\text{m}\text{/}\text{s}$ hits a wall and rebounds with the same speed. FIGURE P11.40 shows the force of the wall on the ball during the collision. What is the value of $F_{\text{max}}$, the maximum value of the contact force during the collision?

FIGURE P11.40

|| INT A 200 g ball is dropped from a height of 2.0 m, bounces on a hard floor, and rebounds to a height of 1.5 m. FIGURE P11.41 shows the impulse received from the floor. What maximum force does the floor exert on the ball?

FIGURE P11.41

|| BIO The flowers of the bunchberry plant open with astonishing force and speed, causing the pollen grains to be ejected out of the flower in a mere 0.30 ms at an acceleration of $2.5\times {10}^4{\text{m/s}}^2$. If the acceleration is constant, what impulse is delivered to a pollen grain with a mass of $1.0\times {10}^{-7}\text{g}?$

| CALC A particle of mass m is at rest at $t=0$. Its momentum for $t>0$ is given by $p_x=6t^2\text{kg m}\text{/}\text{s}$, where t is in s. Find an expression for $F_x\left(t\right)$, the force exerted on the particle as a function of time.

|| Air-track gliders with masses $300\text{g,}400\text{g}$, and $200\text{g}$ are lined up and held in place with lightweight springs compressed between them. All three are released at once. The $200\text{g}$ glider flies off to the right while the $300\text{g}$ glider goes left. Their position-versus-time graphs, as measured by motion detectors, are shown in FIGURE P11.44. What are the direction (right or left) and speed of the 400 g glider that was in the middle?

FIGURE P11.44

|| INT Most geologists believe that the dinosaurs became extinct 65 million years ago when a large comet or asteroid struck the earth, throwing up so much dust that the sun was blocked out for a period of many months. Suppose an asteroid with a diameter of 2.0 km and a mass of $1.0\times {10}^{13}\text{kg}$ hits the earth $(6.0\times {10}^{24}\text{kg})$ with an impact speed of $4.0\times {10}^4\text{m}\text{/}\text{s}$.

1. What is the earth’s recoil speed after such a collision? (Use a reference frame in which the earth was initially at rest.)

2. What percentage is this of the earth’s speed around the sun? The earth orbits the sun at a distance of $1.5\times {10}^{11}\text{m}$.

| A firecracker in a coconut blows the coconut into three pieces. Two pieces of equal mass fly off south and west, perpendicular to each other, at speed $v_0$. The third piece has twice the mass as the other two. What are the speed and direction of the third piece? Give the direction as an angle east of north.

|| BIO Squids rely on jet propulsion to move around. A 1.50 kg squid (including the mass of water inside the squid) drifting at $0.40\text{m/s}$ suddenly ejects 0.100 kg of water to get itself moving at $2.50\text{m/s}$. If drag is ignored over the small interval of time needed to expel the water (the impulse approximation), what is the water’s ejection speed relative to the squid?

|| One billiard ball is shot east at $2.0\text{m}\text{/}\text{s}$. A second, identical billiard ball is shot west at $1.0\text{m}\text{/}\text{s}$. The balls have a glancing collision, not a head-on collision, deflecting the second ball by $90\text{°}$ and sending it north at $1.41\text{m}\text{/}\text{s}$. What are the speed and direction of the first ball after the collision? Give the direction as an angle south of east.

||

1. A bullet of mass $m$ is fired into a block of mass $M$ that is at rest. The block, with the bullet embedded, slides distance $d$ across a horizontal surface. The coefficient of kinetic friction is ${\mu }_{\text{k}}$. Find an expression for the bullet’s speed v_bullet.

2. What is the speed of a $10\text{g}$ bullet that, when fired into a $10\text{kg}$ stationary wood block, causes the block to slide $5.0\text{cm}$ across a wood table?

|| You are part of a search-and-rescue mission that has been called out to look for a lost explorer. You’ve found the missing explorer, but, as FIGURE P11.50 shows, you’re separated from him by a 200-m-high cliff and a 30-m-wide raging river. To save his life, you need to get a 5.0 kg package of emergency supplies across the river. Unfortunately, you can’t throw the package hard enough to make it across. Fortunately, you happen to have a 1.0 kg rocket intended for launching flares. Improvising quickly, you attach a sharpened stick to the front of the rocket, so that it will impale itself into the package of supplies, then fire the rocket at ground level toward the supplies. What minimum speed must the rocket have just before impact in order to save the explorer’s life?

FIGURE P11.50

|| INT An object at rest on a flat, horizontal surface explodes into two fragments, one seven times as massive as the other. The heavier fragment slides 8.2 m before stopping. How far does the lighter fragment slide? Assume that both fragments have the same coefficient of kinetic friction.

|| INT In a ballistics test, a 25 g bullet traveling horizontally at $1200\text{m}\text{/}\text{s}$ goes through a 30-cm-thick 350 kg stationary target and emerges with a speed of $900\text{m}\text{/}\text{s}$. The target is free to slide on a smooth horizontal surface. What is the target’s speed just after the bullet emerges?

|| INT A 1500 kg weather rocket accelerates upward at $10\text{m}\text{/}{\text{s}}^2$. It explodes 2.0 s after liftoff and breaks into two fragments, one twice as massive as the other. Photos reveal that the lighter fragment traveled straight up and reached a maximum height of 530 m. What were the speed and direction of the heavier fragment just after the explosion?

|| Two 500 g blocks of wood are 2.0 m apart on a frictionless table. A 10 g bullet is fired at $400\text{m}\text{/}\text{s}$ toward the blocks. It passes all the way through the first block, then embeds itself in the second block. The speed of the first block immediately afterward is $6.0\text{m}\text{/}\text{s}$. What is the speed of the second block after the bullet stops in it?

|| INT A 100 g toy car sits at the bottom of a $\text{15}°$ ramp. A 200 g toy truck hits the car from behind, causing the car to roll 75 cm up the ramp. How fast was the truck going at the moment of impact? Assume the collision is perfectly elastic.

||| INT A spring with spring constant $50\text{N}\text{/}\text{m}$ is suspended from the ceiling with a 2.0 kg wood block hanging from the bottom. When a 10 g bullet is fired vertically upward into the block, stopping inside, the block begins to oscillate with its highest point 45 cm above its initial level. What was the speed of the bullet?

|| INT FIGURE P11.57 shows a block of mass m sliding along a frictionless surface with speed $v_m$. It collides with a block of mass M that is hanging as a pendulum on a massless rod of length L. The other end of the rod is attached to a frictionless pivot. Find an expression for the $v_m$ that will allow the pendulum to barely go over the top if the collision is (a) perfectly inelastic or (b) perfectly elastic.

FIGURE P11.57

|| Ann (mass 50 kg) is standing at the left end of a 15-m-long, 500 kg cart that has frictionless wheels and rolls on a frictionless track. Initially both Ann and the cart are at rest. Suddenly, Ann starts running along the cart at a speed of $5.0\text{m}\text{/}\text{s}$ relative to the cart. How far will Ann have run relative to the ground when she reaches the right end of the cart?

|| The stoplight had just changed and a 2000 kg Cadillac had entered the intersection, heading north at $3.0\text{m}\text{/}\text{s}$, when it was struck by a 1000 kg eastbound Volkswagen. The cars stuck together and slid to a halt, leaving skid marks angled $35\text{°}$ north of east. How fast was the Volkswagen going just before the impact?

|| CALC Force $F_x=(10\text{N) sin}(2\pi t\text{/}4.0\text{s})$ is exerted on a 250 g particle during the interval $0\text{s}\le t\le 2.0\text{s}$. If the particle starts from rest, what is its speed at $t=2.0\text{s}?$

|| CALC A 500 g particle has velocity $v_x=-5.0\text{m}\text{/}\text{s}$ at $t=-2\text{s}$. Force $F_x=(4-t^2)\text{N}$, where t is in s, is exerted on the particle between $t=-2\text{s}$ and $t=2\text{s}$. This force increases from 0 N at $t=-2\text{s}$ to 4 N at $t=0\text{s}$ and then back to 0 N at $t=2\text{s}$. What is the particle’s velocity at $t=2\text{s}?$

|| A 30 ton rail car and a 90 ton rail car, initially at rest, are connected together with a giant but massless compressed spring between them. When released, the 30 ton car is pushed away at a speed of $4.0\text{m}\text{/}\text{s}$ relative to the 90 ton car. What is the speed of the 30 ton car relative to the ground?

||| A two-stage rocket is traveling at $1200\text{m}\text{/}\text{s}$ with respect to the earth when the first stage runs out of fuel. Explosive bolts release the first stage and push it backward with a speed of $35\text{m}\text{/}\text{s}$ relative to the second stage. The first stage is three times as massive as the second stage. What is the speed of the second stage after the separation?

|| A 100 g ball moving to the right at $4.0\text{m}\text{/}\text{s}$ collides head-on with a 200 g ball that is moving to the left at $3.0\text{m}\text{/}\text{s}$.

1. If the collision is perfectly elastic, what are the speed and direction of each ball after the collision?

2. If the collision is perfectly inelastic, what are the speed and direction of the combined balls after the collision?

|| INT One end of a massless, 30-cm-long spring with spring constant 15 N/m is attached to a 250 g stationary air-track glider; the other end is attached to the track. A 500 g glider hits and sticks to the 250 g glider, compressing the spring to a minimum length of 22 cm. What was the speed of the 500 g glider just before impact?

||| INT Consider a partially elastic collision in which ball A of mass m with initial velocity $(v_{\text{i}x}){}_{\text{A}}$ collides with stationary ball B, also of mass m, and in which $\frac{1}{4}$ of the mechanical energy is dissipated as thermal energy. Find expressions for the final velocities of each ball.

Hint: Mathematically there are two solutions; however, one of them is physically impossible.

||| Most collisions are partially elastic, which means that the objects bounce apart but with some of the mechanical energy dissipated as thermal energy. A partially elastic collision is characterized by a coefficient of restitution r. The analysis of a partially elastic collision can be divided into four steps:

• First, the colliding objects compress, like springs. At the end of the compression phase, the two objects are moving with a common intermediate velocity $v_{\text{int}}$. If the collision ended here, it would be a perfectly inelastic collision. You can calculate $v_{\text{int}}$ and then the momentum change of each object.

• During the compression, each object experiences an impulse $J_{\text{compress}}$ due to the other. You can use the momentum principle to calculate $J_{\text{compress}}$ for each object.

• Second, the objects expand. During expansion, each experiences an impulse $J_{\text{expand}}=r{J}_{\text{compress}}$, where r is the coefficient of restitution. It characterizes how well the objects are restored after deformation. A perfectly inelastic collision has $r=0$, while a perfectly elastic collision has $r=1$.

• Another application of the momentum principle gives the post-collision momentum, and thus the final velocity, for each object.

A 100 g ball (ball A) traveling to the right at $30\text{m/s}$ collides with a stationary 200 g ball (ball B). The materials are such that the coefficient of restitution is 0.25. What is each ball’s final velocity?

|| The nucleus of the polonium isotope ${}^{214}\text{Po}$ (mass 214 u) is radioactive and decays by emitting an alpha particle (a helium nucleus with mass 4 u). Laboratory experiments measure the speed of the alpha particle to be $1.92\times {10}^7\text{m}\text{/}\text{s}$. Assuming the polonium nucleus was initially at rest, what is the recoil speed of the nucleus that remains after the decay?

|| A neutron is an electrically neutral subatomic particle with a mass just slightly greater than that of a proton. A free neutron is radioactive and decays after a few minutes into other subatomic particles. In one experiment, a neutron at rest was observed to decay into a proton $(\text{mass}1.67\times {10}^{-27}\text{kg})$ and an electron $(\text{mass}9.11\times {10}^{-31}\text{kg})$. The proton and electron were shot out back-to-back. The proton speed was measured to be $1.0\times {10}^5\text{m}\text{/}\text{s}$ and the electron speed was $3.0\times {10}^7\text{m}\text{/}\text{s}$. No other decay products were detected.

1. Did momentum seem to be conserved in the decay of this neutron?

   NOTE   Experiments such as this were first performed in the 1930s and seemed to indicate a failure of the law of conservation of momentum. In 1933, Wolfgang Pauli postulated that the neutron might have a third decay product that is virtually impossible to detect. Even so, it can carry away just enough momentum to keep the total momentum conserved. This proposed particle was named the neutrino, meaning “little neutral one.” Neutrinos were, indeed, discovered nearly 20 years later.

2. If a neutrino was emitted in the above neutron decay, in which direction did it travel? Explain your reasoning.

3. How much momentum did this neutrino “carry away” with it?

|| A 45 g projectile explodes into three pieces: a 20 g piece with velocity $25î\text{m}\text{/}\text{s}$, a 15 g piece with velocity $-10î+10ĵ\text{m}\text{/}\text{s}$, and a 10 g piece with velocity $-15î-20ĵ\text{m}\text{/}\text{s}$. What was the projectile’s velocity just before the explosion?

|| A white ball traveling at $2.0\text{m/s}$ hits an equal-mass red ball at rest. The white ball is deflected by $\text{25}°$ and slowed to $1.5\text{m/s}$.

1. What are the final speed and direction of the red ball? Give the direction as an angle from the initial direction of the white ball.

2. What percentage of the initial mechanical energy is lost in the collision?

|| FIGURE P11.72 shows a collision between three balls of clay. The three hit simultaneously and stick together. What are the speed and direction of the resulting blob of clay?

FIGURE P11.72

|| A 2100 kg truck is traveling east through an intersection at $2.0\text{m}\text{/}\text{s}$ when it is hit simultaneously from the side and the rear. (Some people have all the luck!) One car is a 1200 kg compact traveling north at $5.0\text{m}\text{/}\text{s}$. The other is a 1500 kg midsize traveling east at $10\text{m}\text{/}\text{s}$. The three vehicles become entangled and slide as one body. What are their speed and direction just after the collision?

|| A rocket in deep space has an empty mass of 150 kg and exhausts the hot gases of burned fuel at $2500\text{m/s}$. It is loaded with 600 kg of fuel, which it burns in 30 s. What is the rocket’s speed 10 s, 20 s, and 30 s after launch?

||

1. To understand why rockets often have multiple stages, first consider a single-stage rocket with an empty mass of 200 kg, 800 kg of fuel, and a $2000\text{m/s}$ exhaust speed. If fired in deep space, what is the rocket’s maximum speed?

2. Now divide the rocket into two stages, each with an empty mass of 100 kg, 400 kg of fuel, and a $2000\text{m/s}$ exhaust speed. The first stage is released after it runs out of fuel. What is the top speed of the second stage? You’ll need to consider how the equation for $v_{\text{max}}$ should be altered when a rocket is not starting from rest.

$(0.10\text{kg})\left(40\text{m}\text{/}\text{s}\right)-\left(0.10\text{kg}\right)(-30\text{m}\text{/}\text{s})=\frac{1}{2}\left(1400\text{N}\right)\text{Δ}t$

$(600\text{g})\left(4.0\text{m}\text{/}\text{s}\right)=\left(400\text{g}\right)\left(3.0\text{m}\text{/}\text{s}\right)+\left(200\text{g}\right)(v_{\text{i}x}){}_2$

$(3000\text{kg})v_{\text{f}x}=(2000\text{kg})\left(5.0\text{m}\text{/}\text{s}\right)+\left(1000\text{kg}\right)(-4.0\text{m}\text{/}\text{s})$

$\begin{array}{l}(0.10\text{kg}+0.20\text{kg})v_{1x}=(0.10\text{kg})(3.0\text{m}\text{/}\text{s})\hfill \\ \frac{1}{2}(0.30\text{kg})(0{\text{m/s)}}^2+\frac{1}{2}(3.0\text{N}\text{/}\text{m}){(\text{Δ}{x}_2)}^2\hfill \\ =\frac{1}{2}(0.30\text{kg}){\left(v_{1x}\right)}^2+\frac{1}{2}(3.0\text{N/m})(0{\text{m)}}^2\hfill \end{array}$

||| A spaceship of mass $2.0\times {10}^6\text{kg}$ is cruising at a speed of $5.0\times {10}^6\text{m}\text{/}\text{s}$ when the antimatter reactor fails, blowing the ship into three pieces. One section, having a mass of $5.0\times {10}^5\text{kg}$, is blown straight backward with a speed of $2.0\times {10}^6\text{m}\text{/}\text{s}$. A second piece, with mass $8.0\times {10}^5\text{kg}$, continues forward at $1.0\times {10}^6\text{m}\text{/}\text{s}$. What are the direction and speed of the third piece?

||| A 1000 kg cart is rolling to the right at $5.0\text{m}\text{/}\text{s}$. A 70 kg man is standing on the right end of the cart. What is the speed of the cart if the man suddenly starts running to the left with a speed of $10\text{m}\text{/}\text{s}$ relative to the cart?

||| A 20 kg wood ball hangs from a 2.0-m-long wire. The maximum tension the wire can withstand without breaking is 400 N. A 1.0 kg projectile traveling horizontally hits and embeds itself in the wood ball. What is the greatest speed this projectile can have without causing the wire to break?

||| A 20 g ball is fired horizontally with speed $v_0$ toward a 100 g ball hanging motionless from a 1.0-m-long string. The balls undergo a head-on, perfectly elastic collision, after which the 100 g ball swings out to a maximum angle ${\theta }_{\text{max}}=50\text{°}$. What was $v_0\text{?}$

||| The air-track carts in FIGURE CP11.84 are sliding to the right at $1.0\text{m}\text{/}\text{s}$. The spring between them has a spring constant of $120\text{N}\text{/}\text{m}$ and is compressed 4.0 cm. The carts slide past a flame that burns through the string holding them together. Afterward, what are the speed and direction of each cart?

FIGURE CP11.84

||| CALC Section 11.6 found an equation for $v_{\text{max}}$ of a rocket fired in deep space. What is $v_{\text{max}}$ for a rocket fired vertically from the surface of an airless planet with free-fall acceleration g? Referring to Section 11.6, you can write an equation for $\text{Δ}{P}_y$, the change of momentum in the vertical direction, in terms of $dm$ and $d{v}_y$. $\text{Δ}{P}_y$ is no longer zero because now gravity delivers an impulse. Rewrite the momentum equation by including the impulse due to gravity during the time $dt$ during which the mass changes by $dm$. Pay attention to signs! Your equation will have three differentials, but two are related through the fuel burn rate R. Use this relationship—again pay attention to signs; m is decreasing—to write your equation in terms of $dm$ and $d{v}_y$. Then integrate to find a modified expression for $v_{\text{max}}$ at the instant all the fuel has been burned.

1. What is $v_{\text{max}}$ for a vertical launch from an airless planet? Your answer will be in terms of $m_{\text{R}}$, the empty rocket mass; $m_{\text{F}0}$, the initial fuel mass; $v_{\text{ex}}$, the exhaust speed; R, the fuel burn rate; and g.

2. A rocket with a total mass of 330,000 kg when fully loaded burns all 280,000 kg of fuel in 250 s. The engines generate 4.1 MN of thrust. What is this rocket’s speed at the instant all the fuel has been burned if it is launched in deep space? If it is launched vertically from the earth?