00:01
For this problem on the topic of particle physics, we want to consider a collision in which a stationary particle with mass big m is bombarded by a particle with mass little m and speed v0 and total energy em.
00:13
We want to first use the lorentz transformation to rewrite the velocities v for both particles for little m and big m in terms of the speed of the center of momentum.
00:24
We then want to use the fact that the total momentum in the center of momentum frame is zero and then finally expression in terms of the masses and v0 for the speed of the center of momentum.
00:37
Lastly, we want to combine the results from a and b to find the total energy in the center of momentum frame.
00:45
So firstly, for part a, for mass little m, we have vx equal to vx prime plus u divided by 1 plus u, vx, prime divided by c squared and we have u to be minus vc m v prime is equal to v0 and so v little m is equal to v0 minus vcm divided by one minus vcm over c squared similarly for mass capital m we have u equal to minus vc m and v prime equal to zero so the speed vm is equal to minus vcm.
02:09
Now for part b the condition for no net momentum in the center of mass frame is that m little m times gamma little m times v little m plus big m gamma for big m times v m must equal to zero.
02:28
Now gamma little m and gamma big m correspond to the velocities found in part a.
02:33
And so doing the algebra, we get this to be beta little m, gamma little m equal to beta naught minus beta prime times gamma naught, gamma big m...