# Special Relativity GRE question (Where is my method flawed?)

1. Sep 5, 2013

### PsychonautQQ

1. The problem statement, all variables and given/known data
http://grephysics.net/ans/9277/70

I'm not interested in how you get the right answer, I'm really just curious if somebody could show me where my thought process was flawed.

Total Energy = E = 100mc^2
Rest Energy = Mc^2
Total Energy = Kinetic Energy + Rest Energy
100mc^2 = p^2/2m + mc^2
99mc^2 = p^2/2m
198(mc)^2 = p^2
sqrt(198)mc = P
and yet the correct answer is
10^4(mc)

where is my method flawed?

2. Sep 5, 2013

### phyzguy

The formula KE = p^2/(2m) and TE = p^2/(2m) + mc^2 are non-relativistic approximations, and are only valid when pc << mc^2. They come about from taking the full relativistic equation and expanding the square root, keeping only the leading term, as follows:
$$E^2 = p^2 c^2 + m^2 c^4$$
$$E = \sqrt{p^2 c^2 + m^2 c^4} = mc^2 \sqrt{1+\frac{p^2 c^2}{m^2 c^4}}$$
$$E \approx mc^2 (1+\frac{p^2 c^2}{2m^2 c^4}) = mc^2 + \frac{p^2}{2m}$$

3. Sep 5, 2013

4. Sep 5, 2013

5. Sep 6, 2013

### phyzguy

Last edited: Sep 6, 2013