A particle of rest energy mc2 is moving with speed v in the...

A particle of rest energy mc^2 is moving with speed v in the positive x direction. The particle decays into two particles, each of rest energy 140 MeV. One particle, with kinetic energy 282 MeV, moves in the positive x direction, and the other particle, with kinetic energy 25 MeV, moves in the negative x direction. Find the rest energy of the original particle and its speed.

Transcript

00:01 Let's say we have a particle with a rest energy of just mc squared.

00:05 We don't know what it is.

00:07 And let's call it e -0.

00:12 And it is moving to the right with a speed of v, which we're not told what that is.

00:19 And it decays into two particles, one of them which moves to the left, one of them which moves to the right.

00:26 The particle moving to the left has a kinetic energy.

00:30 We'll call this k of 25 mv and the particle moving to the right has a kinetic energy of 282 m eve and let's see what else are we told that's about it and we want to know um or each particle has a rest energy of 140 m eve okay so given all that we want to find what is the uh mass or original sorry, the rest energy of the particle and its speed.

01:05 So for momentum conservation, we know that the total energy of the particle is going to be the square root of its momentum squared times the speed of light squared, plus its rest energy squared.

01:18 And so our task is to first figure out what is this value.

01:22 We can figure out the energy, because from energy conservation, that's kind of straightforward.

01:28 So let's call this particle moving to the left particle one.

01:32 And the other one particle two so we have the the energy of particle one is going to be its rest energy which is 140 mv plus the kinetic energy so 25 mav right and so that's a pretty straightforward 165 and then for particle two uh kind of the same thing we're going to have 140 m evs plus 282 mav and so that gives us something like 422 all right and so from those two facts we can find the total energy because we know it's just going to be the sum of those things so like the the total energy which we'll just call e is going to be e1 plus e2 and so this is going to be something like 587 m ev and so we have one of the variables we just need to find the momentum now so for particle one we know that the momentum uh like times the speed of light you know we can solve if we take this equation that we wrote earlier and solve for the momentum we can see the momentum is going to be um like m squared c to the fourth minus e squared um sorry i've got this backwards e squared minus m squared c to the fourth so uh for particle one which is moving to the left, we're going to have a negative momentum, so it's just important to keep that in mind...

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