Relativistic dynamics of high energy particles

[jianxu]

Homework Statement

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?

Homework Equations

conservation of energy
length contraction
time dilation
lorentz transformations

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

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 ^_^

 

[jianxu]

someone please help