Question

Derive the final velocity equations in a 1D elastic two body collision.

Answer 1

Collisions in 1-dimension

Figure 54: A collision in 1-dimension.

Consider two objects of mass and , respectively, which are free to move in 1-dimension. Suppose that these two objects collide. Suppose, further, that both objects are subject to zero net force when they are not in contact with one another. This situation is illustrated in Fig. 54.

Both before and after the collision, the two objects move with constant velocity. Let and be the velocities of the first and second objects, respectively, before the collision. Likewise, let and be the velocities of the first and second objects, respectively, after the collision. During the collision itself, the first object exerts a large transitory force on the second, whereas the second object exerts an equal and opposite force on the first. In fact, we can model the collision as equal and opposite impulses given to the two objects at the instant in time when they come together.

We are clearly considering a system in which there is zero net external force (the forces associated with the collision are internal in nature). Hence, the total momentum of the system is a conserved quantity. Equating the total momenta before and after the collision, we obtain

(217)

This equation is valid for any 1-dimensional collision, irrespective its nature. Note that, assuming we know the masses of the colliding objects, the above equation only fully describes the collision if we are given the initial velocities of both objects, and the final velocity of at least one of the objects. (Alternatively, we could be given both final velocities and only one of the initial velocities.)

There are many different types of collision. An elastic collision is one in which the total kinetic energy of the two colliding objects is the same before and after the collision. Thus, for an elastic collision we can write

(218)

in addition to Eq. (217). Hence, in this case, the collision is fully specified once we are given the two initial velocities of the colliding objects. (Alternatively, we could be given the two final velocities.)

The majority of collisions occurring in real life are not elastic in nature. Some fraction of the initial kinetic energy of the colliding objects is usually converted into some other form of energy--generally heat energy, or energy associated with the mechanical deformation of the objects--during the collision. Such collisions are termed inelastic. For instance, a large fraction of the initial kinetic energy of a typical automobile accident is converted into mechanical energy of deformation of the two vehicles. Inelastic collisions also occur during squash/racquetball/handball games: in each case, the ball becomes warm to the touch after a long game, because some fraction of the ball's kinetic energy of collision with the walls of the court has been converted into heat energy. Equation (217) remains valid for inelastic collisions--however, Eq. (218) is invalid. Thus, generally speaking, an inelastic collision is only fully characterized when we are given the initial velocities of both objects, and the final velocity of at least one of the objects. There is, however, a special case of an inelastic collision--called a totally inelastic collision--which is fully characterized once we are given the initial velocities of the colliding objects. In a totally inelastic collision, the two objects stick together after the collision, so that .

Let us, now, consider elastic collisions in more detail. Suppose that we transform to a frame of reference which co-moves with the centre of mass of the system. The motion of a multi-component system often looks particularly simple when viewed in such a frame. Since the system is subject to zero net external force, the velocity of the centre of mass is invariant, and is given by

(219)

An object which possesses a velocity in our original frame of reference--henceforth, termed the laboratory frame--possesses a velocity in the centre of mass frame. It is easily demonstrated that

			(220)
			(221)
			(222)