# High-Energy Approximation

1. Jan 16, 2017

### Ken Miller

1. The problem statement, all variables and given/known data
Show that, for an extremely relativistic particle, the particle speed u differs from the speed of light c by
$$c - u = (\frac {c} {2}) (\frac {m_0 c^2} {E} )^2,$$ in which $E$ is the total energy.

2. Relevant equations
I'm not sure what equations are relevant. This problem was listed at the end of a chapter that included:
$$m = \gamma m_0,$$
$$p = \gamma m_0 u,$$
$$E = m_0 c^2 + K,$$
$$E^2 = (pc)^2 + (m_0 c^2)^2,$$
$$dE/dp = u = \frac {pc^2} {E}.$$

3. The attempt at a solution

I have tried to combine/manipulate the above equations into the desired expression, or something similar that I could then use a high-speed approximation on, but I've had no luck. A hint to get me going would be appreciated.

2. Jan 16, 2017

### Orodruin

Staff Emeritus
Can you show us what you tried instead of saying "a few things"?

Clearly, you will not get the expression directly since it is a high speed approximation. You will need to extract u from somewhere and then use that some quantity is much smaller than another quantity in the relevant limit.