# Homework Help: Mass of a composite particle using relativity.

1. May 22, 2012

### lukeharvey

1. The problem statement, all variables and given/known data

A photon collides with a stationary particle of rest mass m0 and is absorbed.

Find the mass and velocity of the composite particle

2. Relevant equations

Conservation of energy:
E + m0 * c^2 = \gamma * mt *c^2 where mt is the total mass

Conservation of momentum:
E/c = \gamma * mt * v

3. The attempt at a solution
I used the attachment as a diagram for the two different frames.

I then solved for the velocity first and found that:

v= E*c/(E+m0*c^2)

I then solved for the mass (mt) and found:

mt = E+m0*c^2/(\gamma*c^2)

However i am not sure if the solution for the total mass is correct because it involves \gamma. If it is correct i was wondering if there was any way to simplify equation, and maybe substitute an expression in for \gamma, so it is not in the equation?

Thanks

#### Attached Files:

• ###### Relativity.png
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2. May 22, 2012

### gabbagabbahey

Why not plug your solution for $v$ into the definition of $\gamma$ and simplify?

3. May 22, 2012

### lukeharvey

I've tried this but it didnt seem to simplify and when i put it into the equation i found it made it even worse. Any ideas?

4. May 22, 2012

### gabbagabbahey

It simplifies fairly nicely for me.... why don't you show me where you are getting stuck?

5. May 22, 2012

### lukeharvey

Im not the best with latex but i can show you the equation i got on here if you can understand it

So i got gamma = 1/(sqrt(1-((((E*c)/(E+m*c^2))^2)/c^2)))

6. May 22, 2012

### gabbagabbahey

The page won't load properly because of the ascii representations in the address, but I get

$$\frac{1}{\gamma}= \sqrt{1-\frac{v^2}{c^2}} =\frac{\sqrt{m_0^2c^4+2Em_0c^2}}{E+m_0c^2}$$

which makes $m$ come out rather nicely as

$$m=\frac{\sqrt{m_0^2c^4+2Em_0c^2}}{c^2}=m_0\sqrt{1+\frac{2E}{m_0c^2}}$$

7. May 22, 2012

### lukeharvey

But i seem to get:

$\sqrt{1-\frac{E^{2}}{E^{2}+M_{0}^{2}c^{2}}}$

But now im stuck thanks for all the help

8. May 22, 2012

### gabbagabbahey

You are probably expanding $v^2$ incorrectly, remember

$$(E+m_0c^2)^2\neq E^2+m_0^2c^4$$

9. May 22, 2012

### lukeharvey

Ah its actually been one of those days, how can i not expand properly? Thanks a lot for the help :)

10. May 22, 2012

### Morgoth

why not going to a system where the two initial particles move with equal and antiparallel spacial mommenta?
That way you can easily find the mass of the particle afterwards by asking the square of the 4mommentum to be invariant, because the M1 in that system will be at rest. "CM" system.

Then by a lorentz transformation I think, you will be able to find that system from your initial one, and so the mommentum of M1 as well at the "LAB" system.

Just a general idea, I am too tired to try it out myself right now, to see if it's simpler or not. I generally like more the invariance of 4mommentum during collisions rather than any other method, which would make me get lost with velocities etc...