# Relativistic dynamics of high energy particles

1. Sep 19, 2008

### jianxu

1. The problem statement, all variables and given/known data
a) a photon of energy E = 2m0c2 hits a resting particle of rest mass m0 and is absorbed by it. What is the subsequent speed of the new particle after absorption

b)Protons have been observed with energies of up to 1021 eV. How thick does Earth appear to the proton?

c) how old does earth appear to the proton?

d)Our galaxy has a diameter of about 105 light years. How long does such a proton take to traverse it in our perception and in the proper time of the proton?

2. Relevant equations
conservation of energy
length contraction
time dilation
lorentz transformations

E total = (m^2)/$$\sqrt{(1+\frac{u^2}{c^2}}$$

3. The attempt at a solution
for part a, I said that energy must be conserved therefore I get 2m0c^2(energy of moving photon) + 2m0c^2(rest mass of stationary particle) = (mc^2)/$$\sqrt{(1+\frac{u^2}{c^2}}$$ and at this point I solved for u and got:
u = $$\sqrt{c^2(1-\frac{m^2}{9m0^2}}$$

I'm not sure if it's right so I wanted to check if it's correct or not

for part b:
I use the following: E' = $$\gamma$$(E - v*p)
E = (mc^2)/$$\sqrt{(1+\frac{u^2}{c^2}}$$
p = (m*u)/$$\sqrt{(1+\frac{u^2}{c^2}}$$

I'm not sure if I'm doing this right so all of this is symbolic
so I solved for u since we know E, so
u = $$\sqrt{c^2(1-\frac{(m*c^2)^2}{E^2}}$$

that allows me to solve for p, then I plug in everything into the transformation equation.
Then I solve for u' using the same method as above and since I also know that L/u = t and so I can use time dilation to find t' which gives us L' = t'*u'?

for part c:
I'm not sure how to solve for this, I guess the age of the Earth in the proton's frame is just however long the proton's life is?

for part d:
Since we know the energy and the speed in both frames can we just apply L/u = t to find the time it takes to traverse the galaxy?

Any suggestions will be welcome, and thanks for taking your time to read/help ^_^

2. Sep 21, 2008