# How much energy a photon has to have the momentum of a 10MeV proton

1. Sep 8, 2009

### ckp

How much energy must a photon have if it is to have the same momentum of a 10-MeV proton?

I am not sure how to go about starting this one. Can someone help me out?

2. Sep 8, 2009

### JesseM

Good general formula to know in SR:

$$E^2 = m^2 c^4 + p^2 c^2$$

Where E is the energy, m is the rest mass, and p is the relativistic momentum. You can see that for p=0 this reduces to E=mc^2. And for a photon, the rest mass m is zero, so this reduces to E=pc.

Meanwhile, for a particle with nonzero rest mass, the relativistic momentum is given by $$p = \gamma mv$$, where v is the velocity and $$\gamma = 1/\sqrt{1 - v^2/c^2}$$

If you know a proton's rest mass m and energy E, you should be able to use these formulas to find its momentum...

3. Sep 8, 2009

### ckp

so E^2 = m^2c^4+p^2c^2

my E^2 is less than my m^2c^4(the 10MeV is less than the mc^2 for a proton) so my p^2c^2 turns out negative. What am I doing wrong?

4. Sep 8, 2009

anyone?

5. Sep 8, 2009

### rl.bhat

First of all find the momentum of proton having energy 10-MeV, using the formula
p = sqrt(2mE) where m is the mass of the proton. Then using E = pc find the energy of the photon.

6. Sep 8, 2009

### ckp

Where did you get p = sqrt(2mE)?

7. Sep 8, 2009

### rl.bhat

p = mv
p^2 = m^2v^2
= 2*m*1/2*m*v^2
= 2*m*E
So p = sqrt(2mE)

8. Sep 9, 2009

### JesseM

Maybe 10 MeV refers to the kinetic energy, which is just the total energy minus the rest mass energy? I'm not sure what the convention when talking about high-energy particles. But it works out that $$E^2 = m^2 c^4 + p^2 c^2$$ is equivalent to $$E = \gamma mc^2$$, so the kinetic energy is $$KE = (\gamma - 1) mc^2$$.