When Momentum Is and Is Not Conserved

Newton's second law in its real form says F_net = dp/dt. Zero net external force means zero change in total momentum. The catch is that "external" only makes sense once you've drawn a boundary around a system. Include both colliding carts and exclude the floor, and the collision force between them is internal, it cancels by Newton's third law, and momentum is conserved during that collision. Leave something out of your system, like the floor or the Earth, and any force from that outside object becomes an external force capable of changing your system's total momentum.

This is why the same physical event can look conserved or not conserved depending entirely on where you draw the line. A force doesn't vanish when you ignore it, it just gets relabeled as external and starts appearing in dp/dt = F_net. Over a short collision time, external forces like friction or gravity deliver only a small impulse compared to the huge internal collision forces, so momentum looks conserved. Stretch out the time window and that small external force keeps acting, and its accumulated impulse, force times time, steadily changes the total momentum.

This connects directly to impulse as the time integral of force: J = integral of F dt. Whether momentum is conserved is really a question of whether the external impulse on your chosen system is negligible over the time interval you care about. Widening the system, say to include the Earth in a falling-ball problem, pulls the force that used to be external, gravity from the Earth, inside the system boundary, making it internal and restoring conservation for the ball-plus-Earth system.