# 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:

- $\mathbf F = \dfrac {m_0 \mathbf a} {\paren {1 - \dfrac {v^2} {c^2} }^{\tfrac 3 2} }$

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 with respect to time:

\(\ds \map {\frac \d {\d t} } {\frac {\mathbf v} {\sqrt {1 - \dfrac {v^2} {c^2} } } }\) | \(=\) | \(\ds \map {\frac \d {\d v} } {\frac {\mathbf v} {\sqrt {1 - \dfrac {v^2} {c^2} } } } \frac {\d v} {\d t}\) | Chain Rule for Derivatives | |||||||||||

\(\ds \) | \(=\) | \(\ds \mathbf a \paren {\frac {\sqrt {1 - \dfrac {v^2} {c^2} } - \dfrac v 2 \dfrac 1 {\sqrt {1 - \dfrac {v^2} {c^2} } } \dfrac{-2 v} {c^2} } {1 - \dfrac {v^2} {c^2} } }\) | Chain Rule for Derivatives, Quotient Rule, etc. | |||||||||||

\(\ds \) | \(=\) | \(\ds \mathbf a \paren {\frac {c^2 \paren {1 - \dfrac {v^2} {c^2} } + v^2} {c^2 \paren {1 - \dfrac {v^2} {c^2} }^{3/2} } }\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \mathbf a \paren {\frac 1 {\paren {1 - \dfrac {v^2} {c^2} }^{3/2} } }\) |

Thus we arrive at the form:

- $\mathbf F = \dfrac {m_0 \mathbf a} {\paren {1 - \dfrac{v^2} {c^2} }^{\tfrac 3 2} }$

$\blacksquare$

## 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.

## Source of Name

This entry was named for Albert Einstein.

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.7$: A Simple Approach to $E = M c^2$