A particle of mass m0 is given a kinetic energy equal to one-third its rest-mass energy. How fast must the particle be traveling? please include units

Answer 1

Kinetic energy =
Total energy - m0*c^2
= (1/3)*m0*c^2
Total energy = (4/3)m0*c^2
= m0*c^2/sqrt[1 - (v/c)^2]

sqrt[1 - (v/c)^2] = 3/4
1 - (v/c)^2 = 9/16
(v/c)^2 = 7/16
v/c = 0.661

You don't need units in that dimensionless form.