# Deriving the Final Velocities

## The Problem:

A particle of mass m1 and velocity v collides elastically (in one dimension) with a stationary particle of mass m2. What are the velocities of m1 and m2 after the collision?

## A particle of mass m1 and velocity v collides elastically with a particle of mass m2, initially at rest. After the collision, m1 has velocity v1, and m2 has velocity v2. What are v1 and v2? The Solution:

Since this is an isolated system, the total momentum of the two particles is conserved:

Also, since this is an elastic collision, the total kinetic energy of the 2-particle system is conserved:

Multiplying both sides of this equation by 2 gives:

Suppose we solve equation 1 for v2:

and then substitute this result into equation 2:

Expanding and multiplying both sides by m2 in order to clear fractions gives:

Now, gather up like terms of v1:

Notice that equation 4 is a standard quadratic in v1, like Ax2 + Bx + C = 0, where:

So, we can use the quadratic formula () to solve for v1:

Inside the radical, the last term of the discriminant has factors like (a + b)(a - b) = a2 - b2, so:

Now, expand and simplify:

So, there are 2 solutions (of course...). Taking the positive sign in the numerator of equation 5 gives:

Physically, this means that no collision took place - the velocity of m1 was unchanged. That isn't the solution we have come this far to find. Taking the negative sign in the numerator of equation 5 gives:

That's it! Now, to find v2, substitute equation 6 into equation 3:

There it is! Equations 6 and 7 give the velocities of the two particles after the collision.

last update November 12, 2009 by JL Stanbrough