modern physics - relativity
posted by walter .
A photon of energy E collides with a stationary particle of rest mass m0 and is absorbed by it .
(A) what is the velocity of resulting composite particle ?
(B) What is the mass of the resulting composite particle ?
use conservasion of energy / momentum please show working
A) You can find the velocity by cmputing the total energy and momentum. From:
E = gamma m c^2
P = gamma m v
it follows that
p/E = v/c^2
The total energy is:
E + m0 c^2
The total momentum is the momentum of the photon, which is E/c. The total energy and momentum are conserved, so these are the energy and momentum of the particle after the photon has been absorbed. The velocity of that particle is thus:
v = c^2 momentum/energy =
c^2 E/c 1/(E + m0 c^2 ) =
E c/(E + m0 c^2)
The mass follows from the invariant:
E^2 - p^2 c^2
Fora system with eenrgy E and momentum p, this quantity has the same valyue in all frames. If you evaluate this in the rest frame then p = 0 and E = m c^2, so it is E^2 = m^2 c^4. The fact tha the quatity is invariant, means that it has this value in any frame. This means that given the enrgy and momentum of a system, you can compute the mass:
m^2 = E^2/c^4 - p^2/c^2
In this case, you find:
m^2 = (E + m0 c^2)^2/c^4 - E^2/c^4 =
2 m0 E/c^2 + m0^2
So, the mass is:
m = m0 sqrt[1 + 2 E/(m0 c^2)]