A particle of mass m travels along the x-axis with speed
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.
|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.