Elastic Collisions in 1 Dimension

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?

Before diagram
after diagram

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:

equation 1

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

conservation of KE equation

Multiplying both sides of this equation by 2 gives:

Conservation of KE equation

Suppose we solve equation 1 for v2:

equation 3

and then substitute this result into equation 2:

equation 2 expanded

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

equation 3b

Now, gather up like terms of v1:

Equation 4

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

quadratic coefficients

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

v1 = quadratic formula

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

equation 4c

Now, expand and simplify:

Equation 5

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

v1 = v

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:

equation 6

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

solution for v2

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

solution summary

last update November 12, 2009 by JL Stanbrough