Momentum, Impulse, and Collisions

Abstract

We demonstrate that the convervation of momentum, $$m \mathbf {v}$$mv, is a result of integrating Newton’s second law over time. If the integral is zero, that is, if the net force is zero, then we find that $$m \mathbf {v}$$mvdoes not change. But how can that be useful? Didn’t we already know from Newton’s first law that if the net external force is zero, the velocity does not change? It turns out that conservation of momentum is not that useful for a single object, but it is very useful for systems consisting of several objects. For systems of several objects, we will demonstrate that the total momentum is conserved if there are no external forces acting on the system. It does not matter what internal forces are acting, the total momentum is conserved at all times as long as there are no external forces. Conservation of momentum is particularly useful for collisions. During a collision between two objects, the interactions between the objects can be very complicated, and may consist of both conservative and non-conservative forces, but as long as the objects are not affected by any external forces, their total momentum is conserved. We use this to find the velocities of each object after a collision from the velocities before a collision, without finding the motion of each object. Conservation of momentum is more general than the conservation of energy, since it is valid for any internal force, and not only for conservative forces. In order to introduce these concepts, we will start by introducing tranlational momentum, $$\mathbf {p}=m \mathbf {v}$$p=mv. We reformulate Newton’s second law using momentum, and find that the integral of the net forces acting on an object corresponds to the change of momentum. We will then use these concepts to address systems with several objects, with a particular focus on collisions.