Einstein's Law of Motion

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

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:
 * $$\mathbf F = \frac{\mathrm{d}}{\mathrm{d}{t}} \left({m \mathbf v}\right)$$

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

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

Then we perform the differentiation WRT time:

$$ $$ $$ $$

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

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