Einstein's Law of Motion

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Physical Law

The force and acceleration on a body of constant rest mass are related by the equation:

$\mathbf F = \dfrac {m_0 \mathbf a} {\left({1 - \dfrac{v^2}{c^2}}\right)^{\tfrac 3 2}}$

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:

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

we substitute Einstein's Mass-Velocity Equation:

$m = \dfrac {m_0} {\sqrt {1 - \dfrac {v^2} {c^2}}}$

to obtain:

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


Then we perform the differentiation with respect to time:

\(\displaystyle \frac{\mathrm d}{\mathrm d t} \left({\frac {\mathbf v}{\sqrt{1 - \dfrac {v^2}{c^2} } } }\right)\) \(=\) \(\displaystyle \frac{\mathrm d}{\mathrm d v} \left({\frac {\mathbf v}{\sqrt{1 - \dfrac {v^2}{c^2} } } }\right) \frac{\mathrm d v}{\mathrm d t}\) Chain Rule
\(\displaystyle \) \(=\) \(\displaystyle \mathbf a \left({\frac {\sqrt{1 - \dfrac {v^2}{c^2} } - \dfrac v 2 \dfrac 1 {\sqrt{1 - \dfrac {v^2}{c^2} } } \dfrac{-2 v}{c^2} } {1 - \dfrac {v^2}{c^2} } }\right)\) Chain Rule, Quotient Rule, etc.
\(\displaystyle \) \(=\) \(\displaystyle \mathbf a \left({\frac {c^2 \left({1 - \dfrac {v^2}{c^2} }\right) + v^2} {c^2 \left({1 - \dfrac {v^2}{c^2} }\right)^{3/2} } }\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \mathbf a \left({\frac 1 {\left({1 - \dfrac {v^2}{c^2} }\right)^{3/2} } }\right)\)


Thus we arrive at the form:

$\mathbf F = \dfrac {m_0 \mathbf a} {\left({1 - \dfrac{v^2}{c^2}}\right)^{\tfrac 3 2}}$

$\blacksquare$


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.


Source of Name

This entry was named for Albert Einstein.


Sources