Collisions in the centre of mass frame

[Lucy Yeats]

I've just found out that in the centre of mass frame, the angle of deflection in a collision is different from in the lab frame.

I vaguely understand why: if the frame you viewed the particles in was also moving but only horizontally it would make their horizontal movement appear to decrease while their vertical movement would stay constant, which would seem to decrease the angle.

I have no idea how you would go about finding angles of deflection in the centre of mass frame. Could someone help me derive/ tell me a formula for doing so?

 

[mathman]

The angle is a free parameter. The issue is transforming between the frames.

 

[Lucy Yeats]

How would I go about transforming between frames?

Thanks for helping! :-)

 

[Philip Wood]

Call one set of axes S. Let another set of axes, S', be co-incident with S. Let S' now move steadily in the +x direction, relative to S. Now suppose there's a particle moving with velocity components u_x, u_y, u_z as described on the S axes. On the S' axes the components will be (u_x-v), u_y, u_z. This is a galilean (non-relativistic) transform.

From the components you can find the direction cosines of the velocity vectors in the two frames. If the particle is moving in, say, just the x and y directions then it's even easier: in S, tanθ = u_y/u_x, whereas in S', tanθ' = u_y/(u_x-v)

 

[Lucy Yeats]

Ah, I think I get it now- thanks.

 

[Philip Wood]

Good! Despite my forgetting to say that v was the velocity of the S' frame relative to the S!