# Elastic and Inelastic Collisions

Of course, collisions come in many forms. We will restrict our attention to two very special kinds.

Inelastic collisions are those in which the two objects stick together.

Consider a situation with mass m1 coming in with initial velocity v1i and striking mass m2 that is at rest (v2i = 0).

The two masses stick together so this is a totally inelastic collision.

The initial momentum is given by

PTot,i = p1i + p2i

PTot,i = m1 v1i

After the collision, the final momentum is given by

PTot,f = (m1 + m2) vf

Since momentum is conserved, this gives

PTot,f = PTot,i

(m1 + m2) vf = m1 v1i

vf = [ m1 / (m1 + m2) ] v1i

Elastic collisions are those in which the Kinetic Energy is also conserved.

Consider two objects that "interact" or collide or have some effect on each other -- in such a way that the total Kinetic Energy is also conserved. This is a totally elastic collision.

PTot,i = PTot,f

m1 v1i + m2 v2i = m1 v1f + m2 v2f

Remember, momentum is a vector! That means that a velocity to the right is positive and a velocity to the left is negative.

Since this is an elastic collision, we also have

K Tot,i = K Tot,f

(1/2) m1 v1i2 + (1/2) m2 v2i 2 = (1/2) m1 v1f2 + (1/2) m2 v2f 2

This provides two equations from which we can solve for the two unknowns, v1f and v2f .

A special case that is interesting to look at is that of an "incoming" object of mass m1 with initial velocity v1i that collides elastically with mass m2 that is at rest. That is v2i = 0.

The equations for the final velocities are

and

Notice what happens when the masses are equal; for m1 = m2:

And notice for m1 > m2:

And notice for m1 < m2:

These elastic collisions are consistent with -- they are predicted by -- these equations:

and

I have some animation for these collisions on my web site for PHY 3050 which I have not yet incorporated into this web site. Click here for that animation.

PHY 3050 has a lighter description of SPLAT - BOING! problems that you might enjoy.