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?
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.