Einstein's Law of Motion

Physical Law
The force and acceleration on a body of constant rest mass are related by the eqn:
 * $\displaystyle \mathbf F = \frac{m_0 \mathbf a}{\left({1 - \dfrac{v^2}{c^2}}\right)^{\tfrac 3 2}}$

where:
 * $\mathbf F$ is the force on the body
 * $\mathbf a$ is the acceleration induced on the body
 * $v$ is the magnitude of the velocity of the body
 * $c$ is the speed of light
 * $m_0$ is the rest mass of the body.

Proof
Into Newton's Second Law of Motion:
 * $\displaystyle \mathbf F = \frac{\mathrm{d}}{\mathrm{d}{t}} \left({m \mathbf v}\right)$

we substitute Einstein's Mass-Velocity Equation:
 * $\displaystyle m = \frac {m_0}{\sqrt{1 - \dfrac {v^2}{c^2}}}$

to obtain:
 * $\displaystyle \mathbf F = \frac{\mathrm{d}}{\mathrm{d}{t}} \left({\frac {m_0 \mathbf v}{\sqrt{1 - \dfrac {v^2}{c^2}}}}\right)$

Then we perform the differentiation WRT time:

Thus we arrive at the form:
 * $\displaystyle \mathbf F = \frac{m_0 \mathbf a}{\left({1 - \dfrac{v^2}{c^2}}\right)^{\tfrac 3 2}}$

Comment
Thus we see that at low velocities (i.e. much less than that of light), the well-known eqn $\mathbf F = m \mathbf a$ holds to a high degree of accuracy.