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