Momentum conservation and collisions

Question

Whenever a collision happens on a rough horizontal surface, because the frictional force will be external to the system, momentum wont be conserved. However, can we consider the friction to be negligible and hence for all practical purposes, momentum should be conserved

Answer 1

In most everyday situations, friction is not negligible. However, frictional forces may be negligibly small compared to other applied forces.

For instance, consider the collision of two pool balls ($m\approx0.2\mathrm{kg}$) on a pool table. Let's say one ball is at least, the other collides with it head on at $v=1\frac{\mathrm m}{\mathrm s}$. And let's say that the collision takes $0.01\mathrm s$, while the balls come to rest due to friction after $3\mathrm s$ (both numbers estimates by myself). Then the forces due to friction are on the order of $F=\frac{\Delta p}{\Delta t}=\frac{0.2\frac{\mathrm{kg m}}{\mathrm s}}{3\mathrm s}\approx0.07\mathrm N$, while the collision forces are on the order of $F=\frac{0.2\frac{\mathrm{kg m}}{\mathrm s}}{0.01\mathrm s}=20\mathrm N$, which is a difference of two orders of magnitude. This means that during the collision, friction can be neglected in the sense that immediately before and immediately after the collision, the loss in total momentum is very small compared to the changes in momenta of the balls due to the collision. And so momentum immediately before and immediately after the collision is approximately the same if we compare the change in momentum to the absolute values of the involved momenta.

Besides this, momentum is still conserved in the sense that if the pool table decelerates a pool ball, then the table itself (and then the ground on which it stands) are slightly accelerated. So counting the total momentum of every involved object will show that momentum is still preserved.