Relativistic Particle Speed Approximation using Total Energy

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For an extremely relativistic particle, the speed u can be approximated by the equation c - u = (c/2)(m_0 c^2/E)^2, where E represents the total energy. Relevant equations include the relationships between mass, momentum, and energy, such as E = m_0 c^2 + K and E^2 = (pc)^2 + (m_0 c^2)^2. The challenge lies in manipulating these equations to derive the desired expression, particularly under high-speed conditions. A hint suggests extracting u from the equations and recognizing that certain quantities become negligible in the relevant limit. Understanding these relationships is crucial for solving the problem effectively.
Ken Miller
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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.
 
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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.
 
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