This is an inelastic collision as the two objects stick together and move together with the same velocity. Inelastic collisions conserve momentum, but they do not conserve kinetic energy.

Consider the law of conservation of momentum.

If both balls are identical then we can say that

Therefore we can state the equation as

Since is the common factor we can remove it from the equation completely.

Since mass factors out of the equation; then it does not matter if the balls increase or decrease in their mass.

Knowns

Unknowns

To solve the problem we must consider three different situations. The first is when the cars are skidding across the ground. While they are skidding the force of friction is what is resisting their motion and therefore doing work on the cars to slow them down to a stop.

The second situation is the collision itself when the first car has an initial speed and the second car is stopped. After the collision both cars are traveling with the different speeds as they skid across the ground at different distances.

The third situation is when the first car initially hits the brakes before the collision. While he is skidding to slow down, the force of friction is resisting the motion and therefore doing work to slow the car down some.

To solve this problem we must work from the end of the collision and work backwards to find out what happened before.

To begin, we need to find the speed that the cars are skidding across the ground after the bumpers have been interlocked.

The work-kinetic energy theorem states that the work done is equal to the change in kinetic energy of the object.

Work is directly related to the force times the displacement of the object.

In this case, the force that is doing the work is friction.

Since the cars are on a level surface the normal force is equal to the force of gravity.

The force of gravity is directly related to the mass and the acceleration due to gravity acting on an object.

When we put all these equations together we get

We also know that kinetic energy is related to the mass and velocity squared.

Therefore our final equation should look like

Notice that the mass falls out of the equation since it is in every term. Also note that the final velocity is 0m/s so this will cancel out as well.

Since both of these terms have a negative we can cancel this out as well.

We can now substitute our variables to determine the velocity of each of the cars just after the crash.

This is the velocity of each of the cars after the collision. We must now consider our second situation of the collision itself. During this collision momentum must be conserved. The law of conservation of momentum states

We can substitute our values for the masses of the cars, the final velocity of the cars after the collision (which we just found) and the initial value of the stopped car.

Now we can solve for our missing variable.

Now we must consider our third situation which is when car A is braking. We can return to our work-kinetic energy theorem equation again as the force of friction is what is slowing the object down.

This time however, the final velocity is not 0m/s as the car A is still moving when he crashes into car B. The final velocity is the velocity that we determined the car was moving with before the collision in the second part of the problem. The masses do still cancel out in this case.

The impulse is equal to the change in momentum. The change in momentum is equal to the mass times the change in velocity.

The yellow ball rebounds higher and therefore has a higher velocity after the rebound. Since it has a higher velocity after the collision, the overall change in momentum is greater. Therefore since the change in momentum is greater, the impulse is higher.

The fastest way to solve a problem like this is with momentum.

Remember that momentum is equal to mass times velocity: . We can rewrite this equation in terms of force.

Using this transformation, we can see that momentum is also equal to force times time.

can also be thought of as .

Expand this equation to include our given values.

Since the crate is not moving at the end, its final velocity is zero. Plug in the given values and solve for the force.

If we were to examine the area under the curve of a constant force applied over a certain amount of time we would have a graph with a straight horizontal line. To find the area of that rectangle we would multiply the base times the height. The base would be the time (number of seconds the force was applied). The height would be the amount of force applied during this time. Force*time is equal to the impulse acting on the object which is equal to the change in momentum of the object.

The impulse is equal to the change in momentum. The change in momentum is equal to the mass times the change in velocity.

The yellow ball rebounds higher and therefore has a higher velocity after the rebound. Since it has a higher velocity after the collision, the overall change in momentum is greater. Therefore since the change in momentum is greater, the impulse is higher.

The fastest way to solve a problem like this is with momentum.

Remember that momentum is equal to mass times velocity: . We can rewrite this equation in terms of force.

Using this transformation, we can see that momentum is also equal to force times time.

can also be thought of as .

Expand this equation to include our given values.

Since the crate is not moving at the end, its final velocity is zero. Plug in the given values and solve for the force.

If we were to examine the area under the curve of a constant force applied over a certain amount of time we would have a graph with a straight horizontal line. To find the area of that rectangle we would multiply the base times the height. The base would be the time (number of seconds the force was applied). The height would be the amount of force applied during this time. Force*time is equal to the impulse acting on the object which is equal to the change in momentum of the object.

Knowns

Unknowns

To solve the problem we must consider three different situations. The first is when the cars are skidding across the ground. While they are skidding the force of friction is what is resisting their motion and therefore doing work on the cars to slow them down to a stop.

The second situation is the collision itself when the first car has an initial speed and the second car is stopped. After the collision both cars are traveling with the different speeds as they skid across the ground at different distances.

The third situation is when the first car initially hits the brakes before the collision. While he is skidding to slow down, the force of friction is resisting the motion and therefore doing work to slow the car down some.

To solve this problem we must work from the end of the collision and work backwards to find out what happened before.

To begin, we need to find the speed that the cars are skidding across the ground after the bumpers have been interlocked.

The work-kinetic energy theorem states that the work done is equal to the change in kinetic energy of the object.

Work is directly related to the force times the displacement of the object.

In this case, the force that is doing the work is friction.

Since the cars are on a level surface the normal force is equal to the force of gravity.

The force of gravity is directly related to the mass and the acceleration due to gravity acting on an object.

When we put all these equations together we get

We also know that kinetic energy is related to the mass and velocity squared.

Therefore our final equation should look like

Notice that the mass falls out of the equation since it is in every term. Also note that the final velocity is 0m/s so this will cancel out as well.

Since both of these terms have a negative we can cancel this out as well.

We can now substitute our variables to determine the velocity of each of the cars just after the crash.

This is the velocity of each of the cars after the collision. We must now consider our second situation of the collision itself. During this collision momentum must be conserved. The law of conservation of momentum states

We can substitute our values for the masses of the cars, the final velocity of the cars after the collision (which we just found) and the initial value of the stopped car.

Now we can solve for our missing variable.

Now we must consider our third situation which is when car A is braking. We can return to our work-kinetic energy theorem equation again as the force of friction is what is slowing the object down.

This time however, the final velocity is not 0m/s as the car A is still moving when he crashes into car B. The final velocity is the velocity that we determined the car was moving with before the collision in the second part of the problem. The masses do still cancel out in this case.