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 $$KE = (\gamma - 1) mc^2$$.