Conservation of Momentum

If we consider Newton's 3rd law, "For every action there is an equal and opposite reaction", it can also be said that forces occur in pairs.  If we consider the 2 skaters, we can see that their momentum is zero, and when they are added together, it equals zero.

The skaters both push and so according to Newtons 3rd law, equal and opposite forces will apply

Now because they exert the force on each other for the came time we can say:

which can be rewritten as:

Rearranging we get:

Which in turn leads to the conclusion:

This makes perfect sense when you remember that momentum is a vector quantity.  Therefore if one object is moving in the opposite direction relative to the other, its value will be negative.

In the scenarios presented to you, it will often involve 2 cars colliding in one form or another.  There are a number of possibilities that may happen as a result of these collisions.

1. The case lock together (in this case we treat them as a single object with mass of the 2 cars combined)
2. The cars are stationary
3. The cars may bounce off each other and move in the opposite direction.