Co-Axial Collision theory without external forces

Consider a body of mass M moving at velocity V approaching a second body. The second body has a mass m and is moving in the same co-axial direction at velocity v. In this format, all symbols dealing with the first body will be denoted by capital letters, i.e., M, V, and all symbols dealing with the second body will be denoted by lower case letters, i.e., m, v. This is a similar format to that followed by Macmillan (1983).

If we arbitrarily choose the body denoted by capital letters to be the "bullet" and the other body as the "target", it follows that in order for the bodies to collide V > v. In contrast to Macmillan’s approach, in which each body is assigned a positive motion as it advances towards the impact, in this theory motion to the right will be deemed positive for both bodies. If m were travelling in the opposite direction it would have a value of –v. However, continuing with Macmillan’s format, the subscript 1 will be used to indicate conditions at the start of impact, and the subscript 2 will represent conditions at the end of impact. Therefore at the start of impact, mass M is moving at velocity V₁ and mass m is moving at velocity v₁. Concentrating on mass M for the moment, when the impact occurs, a force F_c will be generated that will oppose its motion. This force will vary in magnitude throughout the period of the impact.

Newton’s Second Law of Motion defines the general relationship, force equals mass times acceleration, i.e.,

                                                 (1)

When this general relationship is applied to a collision, force F becomes the variable F_c, which will be in effect for the duration of the collision, D t, i.e., from t₁ to t₂. The effect of the impact on mass M may be written as follows:

                               (2)

where I_c is known as the impulse, and has units of newton seconds, or kgm/s. D V is the change of velocity experienced by mass M due to the impulse of the impact. With the sign convention used in this paper, it is defined as:

                                                       (3)

Combining equations (2) and (3):   (4)

Mass m will experience exactly the same force F_c, but in the opposite direction during the collision, as per Newton’s Third Law of Motion and, of course, the force will be felt for exactly the same time, i.e., from t₁ to t₂.

Therefore:                   (5)