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There is, however, a special case of inelastic collisions in which we can predict the outcome. Consider the case in which two particles collide, and actually physically stick together. In this case, called a completely inelastic collision we only need to solve for one final velocity, and the conservation of momentum equation is enough to predict the outcome of the collision. The two particles in a completely inelastic collision must move at the same final velocity, so our linear momentum equation becomes:

m1v1o + m2v2o = m1vf + m2vf    


m1v1o + m2v2o = Mvf    

In this equation M denotes the combined mass of the particles. Thus we can solve for completely inelastic collisions, given the initial conditions.

In studying one-dimensional collisions we are essentially applying the principle of conservation of momentum. The fact that many of these problems are soluble speaks to the importance of this principle. From our understanding of collisions in one dimension, we will move on to the two dimensional case, in which the same principles are applied, but the situations themselves become more complex.