Question

What is the speed, as a fraction of c, of a particle whose momentum is mc?

Short answer

Speed of particle is 0.707c.

Step-by-step solution

Given information

We have given that, particle has momentum mc.

Step 1.

The formula of relativistic momentum is :

$\rho =\frac{mv}{\sqrt{1-{\displaystyle \raisebox{1ex}{$v^2$}\!\left/ \!\raisebox{-1ex}{$c^2$}\right.}}}$

here, m = mass of particle

v = velocity of particle

c = speed of light

$\rho$= relativistic momentum

We have given that the relativistic momentum is mc.

Step 2.

Substituting the values in formula :

$mc=\frac{mv}{\sqrt{1-{\displaystyle \raisebox{1ex}{$v^2$}\!\left/ \!\raisebox{-1ex}{$c^2$}\right.}}}$

$\sqrt{1-\raisebox{1ex}{$v^2$}\!\left/ \!\raisebox{-1ex}{$c^2$}\right.}=\frac{v}{c}$

$1=\frac{2v^2}{c^2}$

$v=\frac{c}{\sqrt{2}}$

$v=0.707c$