A proton has 1836 times the rest mass of an electron At what

A proton has 1836 times the rest mass of an electron. At what speed will an electron have the same kinetic energy as a proton moving at 2.60×102c?

Solution

given
mp = 1836*me

vp = 2.6*10^-2*c = 0.026*c

ve = ?

KEe_rel = KEp_rel

me*c^2/sqrt(1 - (ve/c)^2) = mp*c^2/sqrt(1 - (vp/c)^2

me/sqrt(1 - (ve/c)^2) = mp/sqrt(1 - 0.026^2)

me/sqrt(1 - (ve/c)^2) = me*1836/sqrt(1 - 0.026^2)

1/sqrt(1 - (ve/c)^2) = 1836.62

sqrt(1 - (ve/c)^2) = 1/1836.2

1 - (ve/c)^2 = (1/1836.62)^2

1 - (ve/c)^2 = 2.9645*10^-7

(ve/c)^2 = 1 - 2.9645*10^-7

ve = c*sqrt(1 - 2.9645*10^-7)

= 1.0*c