Einstein's Law of Motion

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

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:
 * $\mathbf F = \map {\dfrac \d {\d t} } {m \mathbf v}$

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

to obtain:
 * $\mathbf F = \map {\dfrac \d {\d t} } {\dfrac {m_0 \mathbf v} {\sqrt {1 - \dfrac {v^2} {c^2} } } }$

Then we perform the differentiation time:

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

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