Use the law of conservation of momentum:

Momentum is the product of velocity and mass:

We can expand the summation for the initial and final conditions:

Note that we are working with an inelastic collision, meaning that the two masses stick together after the collision. Because of this, they will have the same final velocity:

Use the given values to fill in the equation and solve for . Keep in mind that we must designate a positive direction and a negative direction. We will use east as positive and west as negative.

Since the final velocity is positive, we can determine that they train cars are traveling toward the east.

This problem is a conservation of momentum problem. When doing these types of problems, the equation to jump to is:

It is given to us that  is  or ,  is ,  is unknown,  is .

 is  and  is .

With all this information given, the only unknown is .

Plugging everything in, we get:

We must use conservation of momentum to tackle this problem.

We are to find  and , the velocities of the balls after the collision. We know the following for the first ball:

and we know the following for the second ball:

.

After plugging these values into our conservation of momentum equation, it is clear that we can't use this equation alone to find  and ; however, since we are dealing with an elastic collision, we can use the relation below.

This relation can be derived using conservation of momentum and conservation of kinetic energy equations. Remember that kinetic energy is only conserved if the collision is elastic. We can use this relation to eliminate either   or  in our conservation of momentum equation.

Plug this into our conservation of momentum equation.

So now  is eliminated and we can solve for .

This is the speed of the second ball, and it is traveling to the right because of the positive value. Use this positive value to find  by using the relation we found earlier.

This is the speed of the first ball and it is traveling to the left due to the negative sign.

For the 30 kg cart, we know

and for the 46 g cart, we know

.

After the collision, we have .

Use conservation of momentum to solve this problem.

This is a two-dimensional inelastic collision problem and we use conservation of momentum to solve. We know the following.

Male skater: , 

Female skater: ,

First, write down two equations representing conservation of momentum. One equation represents momentum in the x-direction (East-West direction), which is ,and the other gives the momentum in the y-direction (North-South direction), .

Take the y-direction of momentum and divide it by the x-direction momentum.

Simplify.

Plug in the numbers to find the angle

To find the speed at which they move at this angle we can use one of the momentum equations.

Solve for v.

Relevant equations: 

Use conservation of momentum to determine the final velocity of the objects.

Calculate the total initial kinetic energy of the two objects.

Calculate the total final kinetic energy of the two objects.

Find the difference in kinetic energy.

This is the amount of kinetic energy lost during the inelastic collision. Express this amount as a percentage of the initial kinetic energy.

Relevant equations:

Evaluate the impulse based on the mass and change of velocity.

Use the total impulse and time in the second equation.

Solve for the average force.

Relevant equations:

Impulse is defined as change in momentum, so has the same units as momentum. These units can easily be found using the given equations.

To calculate the impulse, know that it is equal to the change in momentum.

Write the impulse equation in terms of mass and velocity.

In our case, the initial velocity of the baseball is  and its final velocity is , where the negative sign indicates that the ball is traveling in the opposite direction. Also, m = 0.15kg. The impulse on the ball can be calculated below.

The negative sign tells that the force on the baseball is opposed to the original momentum.

We need to use conservation of momentum to solve this problem.

Initially, the pickup truck moves at . The load moves together with the truck so the load is also moving at .

We can calculate the truck-load momentum as follows:

Note that this is the sum of the truck's momentum AND the load's momentum. We are looking at is as a whole since they are moving together.

The car moves at . Be careful with signs here! The problem says it moves in the opposite direction. We used a positive velocity for the truck/load, so the velocity of the car should be negative if it is moving in the opposite direction. We can also see from the diagram that the car is moving to the left.

Therefore, we have that the momentum for the car as:

We can find the initial momentum of the system (truck, load, and car) via simple addition:

After the collision, we know that all three objects move together, so they all move with the same velocity. Therefore we can express the momentum of the system after the collision as follows:

 is the velocity of the objects after the collision. This is the sum of the momentum of all three objects; we were able to simplify the equation because they all move with the same velocity.

By conservation of momentum we know that the momentum of the system before the collision is equal to the momentum of the system after the collision.

Use this to solve for the final velocity.