A collision is an isolated event where two or more bodies exert forces on each other, in a relatively short time interval.
During a collision, the involved objects exchange very intense internal forces, called impulsive forces, in a relatively short time interval. These forces vary in a complex manner and cause deformations and changes in velocities of objects. Faced with high internal forces, we can generally neglect the external forces and consider the system, which is formed by objects colliding, as mechanically isolated. Thus, we can apply during a collision the Principle of Movement Conservation to mechanically isolated systems, that is: $$ P_i = P_f,$$ where \(P_i\) and \(P_f\) are, respectively, the initial linear moments, before the collision, and final moment, after a system crash. The following definitions are important:
When two objects collide, there is always a deformation stage. A second stage is also possible, the restitution stage, but it might not happen.
For a given material, we can find experimentally what is the relationship between speed \(v_{after}\) at whitch the particles get away of each other after the collision, and the approach speed \(v_{before}\) before the collision. This relationship is named restitution coefficient \(e\): $$e = \frac{v_{before}}{v_{after}}$$ or $$e = - \frac{v_{i,2}-v_{i,1}}{v_{f,2}-v_{f,1}},$$ where \(v_{i,1}\) and \(v_{i,2}\) are the initial speeds of particles 1 and 2, respectively, and \(v_{f,1}\) and \(v_{f,2}\) are their final speeds.
It is important to note the following points:
Central and Front Collision | Momentum | Kinetic energy | Coefficient of Restitution |
Elastic | \(P_i = P_f\) | \(E_{k_i} = E_{k_f}\) | \(e = 1\) |
Partially Elastic | \(P_i = P_f\) | \(E_{k_i} \gt E_{k_f}\) | \(0 \lt e \lt 1\) |
Inelastic | \(P_i = P_f\) | \(E_{k_i} \gt E_{k_f}\) | \(e = 0\) |