Physics 220: A 2-d collision

A particle of mass m travels along the x-axis with speed v0. It collides elastically with a second particle of mass 2m that is initially at rest. After the collision particle 2 is observed to move at an angle of 30° to the x-axis. Find the velocity of each particle after the collision.

Here's the picture:

Now we can use conservation of energy and momentum to solve for the velocities after the collision. (NB-- "velocity" implies magnitude and direction.) We have drawn the velocity of particle 1 downward so as to have total y-component of momentum equal to zero.
  Before After
Px mv0 mu1cosf + 2mu2cos30°
Py 0 -mu1sinf + 2mu2sin30°
Kinetic energy mv02/2 Mu12/2 + 2mu22/2

Set totals before equal to totals after:


From the second equation, we get u2. Then we can substitute into the other equations.

Now combining these results, we get:

u12(cosf +Ö 3 sinf )2 = u12(1 + 2sin2f)

Now expand out, and use sin2 f + cos2 f = 1 to get:

cosf sinf = 0

Thus either cosf or sinf = 0. But if we take sinf = 0 we cannot satisfy the Py equation, so cosf = 0 and thus f = p /2. Then we get u2 = u1 = v0 /Ö 3.