What is its speed?

Question

A high speed proton has kenetic energy of 898 MeV. What is its speed? (the rest energy of a proton is 939 MeV) The answer is .859c but i am not sure how to solve it.

Answer 1

Rest energy means that if we make a proton move at speed of light its mass will become pure energy. Remember E=mc^2?
So some Lorentz transformations (since that critter is moving pretty fast)

Ke=(k -1) mc^2 where k = 1/sqrt(1 - v^2 / c^2)
k= (Ke + mc^2)/mc^2
and
v^2=[1 - 1/k^2] c^2
v= c sqrt( 1 - 1/k^2)

finally

v= c sqrt[1 - (mc^2 / (Ke + mc^2))^2]

mc^2=939 MeV
Ke= 898 MeV

Just plug the numbers in