Understanding Momentum

Question

I'm trying to learn more about momentum and I'm a little confused. Based on my understanding, in an isolated system, total momentum is conserved in a collision. Today in class the professor went over an example of a car on a ferry driving from one end to the other, in the opposite direction of the ferry. According to him, the total momentum for the system was zero in this case. I understand why total momentum is zero in a collision: because the objects come at rest. But in the example both the car and the ferry are moving (the goal is to find the ferry's new velocity as the car takes off).

So my question is: Is total momentum always zero if two objects are in touch with each other and are applying the same amount of force (but in the opposite direction) to each other? Also, the water applies the same amount of force to the ferry (again in the opposite direction) but how come it's not considered in our "system"?

Answer 1

The rule is that the total momentum of an isolated system is constant. In the example with the car and the ferry, the isolated system is the system consisting of the car and the ferry. The system is isolated because it is assumed that there is no interaction (such as drag) between the ferry and the water.

Then you could imagine that the car and ferry are stationary with respect to each other. Then as the car begins to move and gains momentum in one direction, the ferry gains momentum in the other direction. Conservation of momentum guarantees that the car and ferry momenta add to zero so that the total momentum is zero.

It is important to notice that the rule is not that total momentum is zero, but that total momentum is conserved. So if the car and ferry were initially drifting with some speed, then when the car starts moving across the ferry, the car will gain momentum and the ferry will lose momentum but the total momentum will remain at its initial non-zero value.

Also notice that in a more realistic model, you would include the drag force from the water. Then the momentum of the car/ferry system would not be conserved (which is allowed because it is no longer an isolated system). But if you add in the momentum of the water, you will find that still the total momentum is conserved.

Answer 2

Actually in a collision the force is undetermined until the full details of material properties and geometry is considered. What can be determined is the impulse magnitude. Impulse is the amount of momentum transferred from one body to another during a collision. If you were to plot the contact force as a function of time, it will show a sharp spike during the collision, and the impulse equals the area under the curve. The shorter the duration of the impact the higher the force needs to be for the same impulse. It is a common misunderstanding to try to work out forces and impulses at the same time. Because collisions are set to occur at zero time, any applied force does not have a chance to alter the momentum of the parts.

More on the math behind how to find the impulse magnitude here and here.

To answer your question, whenever two bodies collide they exchange momentum (impulse) and thus the total amount remains the same. Non contact forces during a collision do not affect the momentum because a force has to be applied over a period of time to change the velocity of a part and collisions occur in almost zero time.