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A $\mathrm{100\,g}$ rubber ball is thrown horizontally with speed of $5\ \mathrm{m/s}$ toward a wall. It is initially travelling to the left. It rebounds with no loss of speed. The collision force is shown in the top graph below.

(a) What is the value of maximum force $F_{max}$?

(b) Draw an acceleration vs. time graph for the collision on the middle set of axes below. Make your graph align vertically with the force graph, and provide and appropriate numerical scale on the vertical axes.

(c) Draw a velocity vs. time graph for the collision on the bottom set of axes below. Make your graph align vertically with the acceleration graph, and provide and appropriate numerical scale on the vertical axes.

$1200\, \mathrm{kg}$ car moving at $15\ \mathrm{m/s}$ collides head on with a $2000\, \mathrm{kg}$ truck initially at rest and sticks to the truck after the collision.

(a) What is their velocity just after the collision?

(b) What is the impulse on the car?

(c) If the collision lasts $0.5\, \mathrm{s}$, what is the average deceleration and force of the car?

(d) What fraction of the initial energy is lost in the collision?

A $0.5\ \mathrm{kg}$ cart and a $2.0\, \mathrm{kg}$ car are attached and are rolling forward with a speed of $2.0\ \mathrm{m/s}$. Suddenly a spring-loaded plunger pops out and blows the two carts apart from each other. The smaller mass cart shoots backward at $2.0\, \mathrm{m/s}$.

(a) What are the speed and direction of the $2\, {\rm kg}$ cart?

(b) If the spring constant of the plunger is $25000\, {\rm N/m}$, by how much was the spring initially compressed?

A ball of mass $0.25\ \mathrm{kg}$ is dropped vertically from a height of $1.8\ \mathrm{m}$ and bounces back to the original height. What is the magnitude of the impulse that the floor exerts on the ball during the bounces?

Two blocks, $M=2m$, sit on a horizontal frictionless surface with a compressed massless spring between them. After the spring is released $M$ has velocity $v$. What was the total energy initially stored in the spring?

Heavy wooden block rests on a flat table and a high-speed bullet is fired horizontally into the block, the bullet stopping in it. How far will the block slide before coming to stop? The mass of the bullet is $10.5\,\mathrm{g}$, the mass of the block is $10.5\,\mathrm{kg}$ the bullet's impact speed is $750\ \mathrm{m/s}$, and the coefficient of kinetic friction between the block and the table is $0.220$.

In a pool game, the cue ball, which has an initial speed of $5.0\,\mathrm{m/s}$, makes an elastic collision with the eight ball, which is initially at rest. After the collision, the eight ball moves at an angle of $30{}^\circ $ to the right of the original direction of the cue ball. Assume that the balls have equal mass.

(a) Find the direction of the motion of the cue ball immediately after the collision.

(b) Find the speed of each ball immediately after the collision.

A heavy ball with a mass of $2.0\, \mathrm{kg}$ is sitting on a tabletop. A second, lighter ball is bowled toward the first ball with goal of knocking the heavy ball off the table. The light ball has a mass of $0.25\ \mathrm{kg}$.

(a) When the light ball collides with the heavy ball, it is seen that the heavy ball rolls at a speed of $1.0\ \mathrm{m/s}$ after being hit. If the collision between the two balls is elastic, then how fast was the light ball moving when it struck the heavy ball?

(b) If instead of being elastic collision, the light ball is covered with super-glue so that the two balls stick together, then how fast would the light ball need to be moving when it struck the heavy ball? The final, after collision, speed of the heavy ball is still $1.0\ \mathrm{m/s}$.

Two cars, one a compact car with mass of $1200\,{\rm kg}$ and the other a large pickup truck with mass $3000\, {\rm kg}$, collide head-on at a typical freeway speeds ($70\, {\rm mph}$). The collision is inelastic and two vehicles stuck together after the collision.

(a) What is the speed of the vehicles after the collision?

(b) What are the changes of velocity for the compact car and for the truck?

(c) Which vehicle has a greater change in linear momentum?

A bullet of mass $m=0.01\ \mathrm{kg}$ is fired at a wooden block of mass $M=1.00\ \mathrm{kg}$ as shown. The velocity of the bullet just before it strikes the block is $v_0=850\,\frac{\mathrm{m}}{\mathrm{s}}$. The bullet passes through the block, and rises to a maximum height $h=1200\ \mathrm{m}$.

(a) What is the velocity $v_2$ of the bullet just after it exits the blocks?

(b) What is the $v_1$ the velocity of the block just after the bullet has left it?

(c) What is the kinetic energy that is lost in the collision?

(d) The bullet passes through the block in $3\times {10}^{-5}\,\mathrm{s}$. What is the average force (magnitude and direction) that the block exerts on the bullet?

A $24\ \mathrm{g}$ bullet is shot vertically into an $3\mathrm{kg}$ block. The block lifts upward $8.00\ \mathrm{mm}$. The bullet penetrates the block and comes to rest in it in a time interval of $0.0010\ \mathrm{s}$. Assume the force on the bullet is constant during penetration and that air resistance is negligible. What is the initial kinetic energy of the bullet?

During a collision with a wall, the velocity of a $0.2\ \mathrm{kg}$ ball changes from $20\ \mathrm{m/s\ }$toward the wall to $12\ \mathrm{m/s}$ away from the wall. If the time the ball was in contact with the wall was $60\ \mathrm{ms}$, what was the magnitude of the average force applied to the ball.

Captain Kirk is located $124\ \mathrm{m}$ from the Starship Enterprise and at rest with respect to the Enterprise. Unfortunately, Kirk's $36.6\ \mathrm{kg}$ jetpack has malfunctioned. Kirk in his spacesuit has a mass of $366\ \mathrm{kg}$. The only way for the Kirk to get back to the enterprise is to throw his jetpack out in the opposite direction of the Enterprise. Kirk is able to throw this jetpack away from him at $1.55\ \mathrm{m/s}$. How long (in minutes) will it take Captain Kirk to get back to the Enterprise?

A cannon of mass $M=500\ \mathrm{kg}$, fixed to the ground, shoots a cannon ball of mass $m=20\ \mathrm{kg}$ at a speed of $1000\ \mathrm{m/s}$. Now the stops holding the cannon to the ground are removed, so that it is free to move. The same charge of gunpowder is used and an identical ball fired. What is the speed of the ball?

A lead bullet of mass $m=10.0\ \mathrm{g}$ is travelling with a velocity of $v_0=100\ \mathrm{m/s}$ when it strikes a wooden block. The block has a mass of $M=1.00\ \mathrm{kg}$ and is at rest on the table, as shown in the diagram below. The bullet embeds itself in the block and after the impact they slide together. All the kinetic energy that is lost in the collision is converted into heat. Assume that all this heat goes into heating up the bullet.

(a) What is the speed of the wooden block after the collision?

(b) How much heat is generated as a result of the collision?

(c) By how many degrees does the temperature of the bullet rise after the collision? (The specific heat of the bullet is $128\ \mathrm{J/(kg.{}^\circ\!{C}})$.)

(d) Suppose that the table has a coefficient of kinetic friction of $\mu=0.2$. At what distance will the block stop?

A hockey puck of mass $m=0.35\ \mathrm{kg}$ is sliding without friction on ice, with a velocity of $8.0\ \mathrm{m/s}$ in the $x$ direction. A hockey player hits the puck with his stick, applying a constant force for $0.1\ \mathrm{s}$. As a result of this force, the puck changes direction and subsequently moves off with a velocity of $15.0\ \mathrm{m/s}$ entirely in the $y$ direction.

(a) What are the $x$ and $y$ components of the force?

(b) Give the magnitude of the force and the angle it makes relative to the positive $x$ axis.

A $0.05\ \mathrm{kg}$ hockey puck (labeled $m_2$) is at rest in the center of an air hockey table when a smaller puck with mass $m_1=0.025\ \mathrm{kg}$ moving with velocity $v_{i1}=5\ \mathrm{m/s\ }$collides with it. After collision the pucks move off as shown in the picture.

(a) What is the velocity of each puck after the collision?

(b) Is this collision elastic?

Two equal mass people are symmetrically located at the extreme ends of a uniform $1.5\ \mathrm{m}$ long platform. The combined mass of the people and platform is $150\ \mathrm{kg}$. Additionally one person holds a $6\ \mathrm{kg}$ ball at end $A$ of the platform. The platform sits on a frictionless surface.

(a) What is the $x$ coordinates of the center of the mass of the system.

The person on end A now throws the ball to the other person who catches it right at the opposite end of the platform.

(b) Does the center of mass of the whole system change location?

(c) After the ball is caught, what is the $x$ coordinate of end A of the platform?

A man of mass $M=60\ {\rm kg}$ stands in the center of a platform one of mass $m=40\ {\rm kg}$ that rests on ice. Another identical platform floats at a distance $L=1.5\ {\rm m}$ away. The contact of the two platforms with the ice is frictionless. The man jumps from the first to the second platform, and lands in the center of the second platform.

(a) Locate the center of mass of the two-platform and man system before the man jumps. Use the center of mass of platform one as origin.

(b) What is the velocity $v_{CM}$of the center of mass of the two-platform and man system when the man is midway between the two platforms?

(c) How far apart are the two platforms when the man lands?

An $8\ \mathrm{g}$ bullet is shot into a $4.0\ \mathrm{kg}$ block, at rest on a frictionless surface. The bullet remains lodged in the block. The block moves into a spring and compresses it by $3.0\ \mathrm{cm}$. The force constant of the spring is $1500\ \mathrm{N/m}$.

(a) What is the initial velocity of the bullet?

(b) What is the impulse, due to the spring, during the entire time interval in which block and spring are in contacts?

Category : Momentum and Collision

MOST USEFUL FORMULA IN MOMENTUM AND COLLISION:

Momentum of a particle:

\[\vec p=m\vec v\]

Newton's second law :

\[\Sigma \vec F=\frac{d\vec p}{dt}\]

Kinetic energy of a particle:

\[K=\frac 1 2 mv^2\]

Impulse:

\[\vec J=\Sigma \vec F \left(t_2-t_1\right)\]

Impulse-Momentum theorem:

\[\vec J_{net}=\Sigma \vec F_{net} \Delta t=\Delta P\]

Relative speeds of approach and separation in elastic collision:

\[v_{2f}-v_{1f}=v_{1i}-v_{2i}\]

Center of mass:

\[\vec r_{cm}=\frac {\Sigma m_i \vec r_i}{\Sigma m_i}=\frac{m_1\vec r_1+m_2\vec r_2+\cdots}{m_1+m_2+\cdots}\]

Number Of Questions : 20

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