Knowns

Unknowns

For elastic collisions we know that the initial and final velocities are related by the equation

We also know that the momentum is conserved meaning that

Since we have two missing variables and two equations, we can now solve for one of the variables using a system of equations

Let’s get the final velocity of car A by itself from the first equation

We can now substitute this equation into our momentum equation.

In our new equation we only have one missing variable, so lets substitute in values and solve.

We can now plug this value back into our equation for  

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.

Elastic collisions occur when two objects collide and kinetic energy isn't lost. The objects rebound from each other and kinetic energy and momentum are conserved. Inelastic collisions are said to occur when the two objects remain together after the collision so we are dealing with an elastic collision.

Above, the subscripts 1 and 2 denote puck A and B respectively, and the initial momentum of puck B is zero, so that term is not included in the equation above.

Plug in initial and final velocities and mass:

We know that the cars stick together after the collision, which means that the final velocity will be the same for both of them. Using the formula for conservation of momentum, we can start to set up an equation to solve this problem.

First, we will write the initial momentum.

We know that the second car starts at rest, so this equation can be simplified.

Now we will write out the final momentum. Keep in mind that both cars will have the same velocity!

Set these equations equal to each other and solve to isolate the final velocity.

This is our answer, in terms of the given variables.

Knowns

Unknowns

To solve the problem we must consider two 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 same speed as their bumpers have been locked together.

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 both cars just after the crash.

This is the velocity 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.

This is an example of an inelastic collision, as the two cars stick together after colliding. We can assume momentum is conserved.

To make the equation easier, let's call the first car "1" and the second car "2."

Using conservation of momentum and the equation for momentum, p=mv, we can set up the following equation.

Since the cars stick together, they will have the same final velocity. We know the second car starts at rest, and the velocity of the first car is given. Plug in these values and solve for the final velocity.

The difference between an elastic and an inelastic collision is the loss or conservation of kinetic energy. In an inelastic collision kinetic energy is not conserved, and will change forms into sound, heat, radiation, or some other form. In an elastic collision kinetic energy is conserved and does not change forms.

Remember, total energy and total momentum are conserved regardless of the type of collision; however, while energy cannot be created nor destroyed, it can change forms.

In the answer options, only one choice preserves the total kinetic energy. The resulting bang from the car crash, the flame from the match, the sound of hands clapping, and the gamma radiation during hydrogen fusion are all examples of the conversion of kinetic energy to other forms, making each of these an inelastic collision. Only the neutron fusion described maintains the conservation of kinetic energy, making this an elastic collision.