Collisions in 2-dimensions

Glancing collisions (i.e. not head-on) are more complicated to analyze. Figure 1.1 shows a typical configuration.

FIGURE 1.1 Glancing collision.

Example Write down the conservation of energy and momentum equations for the glancing collision depicted in Fig. 1.1 where the target ball is initially at rest.

Solution Conservation of momentum is

But  In x and y components these are

or

If the collision is elastic we also have conservation of kinetic energy,

These three equations must then be solved for the quantities of interest.

Example A ball of mass m₁ and speed v_1i collides with a stationary target ball of mass m₂, as shown in Fig. 1.1. If the target is scattered at an angle of θ₂ what is the scattering angle θ₁ of the projectile in terms of m₁, m₂, v_1i, θ₂ and v_2f where v_2f is the final speed of the target ?

Solution Conservation of momentum gives

or

The x direction gives

We want to find θ₁. Solve the first and second equations for θ₁ giving

and

giving

(Notice that this result is valid for both elastic and inelastic collisions. We did not use conservation of energy.)