How does conservation of momentum apply practically?

Question

I'm doing AP Physics mechanics and I'm learning about the conservation of momentum and the textbook describes the conditions for it's validity as the net force on the system must be 0. I understand how this works for frictionless systems but I was just curious as to how it applies practically since friction is always present in real life.

I know that you can bypass this by including the source of the force in the system but if you were analysing, say, a real car collision, how would you apply the conservation of momentum to it since the tires have friction? How would you include the road in the boundaries of the system if the road is connected to the entire Earth or would conservation of momentum just not apply at all in this case?

Answer 1

how would you apply the conservation of momentum to it since the tires have friction?

This is a very good question. We know that for any system $$\Delta \vec p =\int_{t_0}^{t_f} \vec F_{net} \ dt$$ For an isolated system $\vec F_{net}=0$ which implies $\Delta \vec p=0$. In other words, as you said, momentum is conserved for an isolated system.

For a non-isolated system we can consider the impulse $$\int_{t_0}^{t_f} \vec F_{net} \ dt=I$$ to be an "error" term from incorrectly applying the conservation of momentum to the non-isolated system.

Notice, as long as $\vec F_{net}$ is finite then as $t_f-t_0=\Delta t \rightarrow 0$ the error term $I \rightarrow 0$ also. Friction is finite. So as long as the duration of the collision is close to $0$ then we can apply the conservation of momentum to the cars during the collision despite the fact that there is friction.

So typically we will consider friction as the cars come towards the collision, we will neglect friction during the almost instantaneous collision and use conservation of momentum, and then we will consider friction again as the cars leave the collision.

Answer 2

It is preferable to take the view that momentum is exactly conserved, always. What you think of as ``not conserving" is just that you neglected an interaction exchanging momentum.

In the case of friction, it is nicer to see it as that the frictional interaction is robbing the object you are looking at of momentum, transferring that momentum to the table (and maybe eventually to the entire Earth).

But we are most often considering the conservation of momentum equations in the instantaneous collision limit. In that case, the collision time is zero, and any finite force (friction in this case) multiplied by zero time of transfer, leads to zero transfer of momentum (impulse), and thus the momentum of the object just before the instantaneous interaction is equal to the same just after the interaction. As such the naïve conservation of momentum is true as long as instantaneous impulses are not happening. (If it were, then we would simply insert these impulses, and then momentum is conserved again.)

It is precisely that there are such conserved quantities that we may always keep track of the exchanges of them.