A particle of mass m_{1} and velocity v collides elastically (in one dimension) with a stationary particle of mass m_{2}. What are the velocities of m_{1} and m_{2} after the collision?


A particle of mass m_{1 }and velocity v collides elastically with a particle of mass m_{2}, initially at rest. 
After the collision, m_{1} has velocity v_{1}, and m_{2} has velocity v_{2}. What are v_{1} and v_{2}? 
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 2particle system is conserved:
Multiplying both sides of this equation by 2 gives:
Suppose we solve equation 1 for v_{2}:
and then substitute this result into equation 2:
Expanding and multiplying both sides by m_{2 }in order to clear fractions gives:
Now, gather up like terms of v_{1}:
Notice that equation 4 is a standard quadratic in v_{1}, like Ax^{2} + Bx + C = 0, where:
So, we can use the quadratic formula () to solve for v_{1}:
Inside the radical, the last term of the discriminant has factors like (a + b)(a  b) = a^{2}  b^{2}, 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 m_{1} 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 v_{2}, substitute equation 6 into equation 3:
There it is! Equations 6 and 7 give the velocities of the two particles after the collision.