Relativistic Particle Speed Approximation using Total Energy

[Ken Miller]

Homework Statement

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.

Homework Equations

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

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.

 

[Orodruin]

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.