Question

Calculate the momentum of a proton whose kinetic energy is 200 MeV.

Answer 1

Given

$E^2 - p^2 c^2 =m^2 c^4$ where E is full energy, p is momentum and m - mass,

and

$E = T+E_0 = T+mc^2$

we obtain

$(T+mc^2)^2 - p^2c^2 = m^2 c^4$

$T^2 +2Tmc^2 + m^2c^4 -p^2c^2 =m^2c^4$

$p^2c^2 = T^2 +2Tmc^2$

$pc = \sqrt{T^2+2Tmc^2}$

Let's calculate all in units of GeV, $T=200\; MeV , mc^2 =938 \; MeV$.

$pc = \sqrt{200^2 + 2\cdot 200 \cdot 938}=\sqrt{415\,200}=644.36 \; MeV$

Answer: p = 644.36 MeV/c.