A particle of mass m travels along the x-axis with speed
v_{0}.
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 | |
P_{x} | mv_{0} | mu_{1}cosf + 2mu_{2}cos30° |
Py | 0 | -mu_{1}sinf + 2mu_{2}sin30° |
Kinetic energy | mv_{0}^{2}/2 | Mu_{1}^{2}/2 + 2mu_{2}^{2}/2 |
Set totals before equal to totals after:
and
From the second equation, we get u_{2}. Then we can substitute into the other equations.
Now combining these results, we get:
u_{1}^{2}(cosf +Ö 3 sinf )^{2} = u_{1}^{2}(1 + 2sin^{2}f)
Now expand out, and use sin^{2} f + cos^{2} f = 1 to get:
cosf sinf = 0
Thus either cosf or sinf = 0. But if we take sinf = 0 we cannot satisfy the P_{y} equation, so cosf = 0 and thus f = p /2. Then we get u_{2} = u_{1} = v_{0} /Ö 3.