Problem 21E

21. The force $F$ acting on a body with mass $m$ and velocity $v$ is the rate of change of momentum: $F=(d / d t)(m v)$. If $m$ is constant, this becomes $F=m a$, where $a=d v / d t$ is the acceleration. But in the theory of relativity the mass of a particle varies with $v$ as follows: $m=m_{0} / \sqrt{1-v^{2} / c^{2}}$, where $m_{0}$ is the mass of the particle at rest and $c$ is the speed of light. Show that

$$ F=\frac{m_{0} a}{\left(1-v^{2} / c^{2}\right)^{3 / 2}} $$


Step-by-Step Solution