Question

At what speed, as a fraction of c, is a particle’s momentum twice its Newtonian value?

Short answer

Speed of particle will be 0.866c.

Step-by-step solution

Given information 

We have given that, particle’s momentum twice its Newtonian value.

Step 1.

The Newtonian momentum is $\rho =mv$

Relativistic momentum of the particle is $\rho =\frac{mv}{\sqrt{1-{\displaystyle \raisebox{1ex}{$v^2$}\!\left/ \!\raisebox{-1ex}{$c^2$}\right.}}}$

As we have given that relativistic momentum is twice of Newtonian momentum, so :

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

Step 2.

$1-\frac{v^2}{c^2}=\frac{1}{4}$

$v=\frac{\sqrt{3}}{2}c$

$v=0.866c$