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title: Einstein's Law of Motion date: 2019-02-24 17:22:28 category: physical tags:

  • physical
  • "law of motion"
  • law
  • montion
  • einstein mathjax: true

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

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:

  • 𝐅 is the force on the body
  • 𝐚 is the acceleration induced on the body
  • 𝑣 is the magnitude of the velocity of the body
  • 𝑐 is the speed of light
  • 𝑚0 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: $$ m = \dfrac {m_0} {\sqrt {1 - \dfrac {v^2} {c^2}}} $$ where:

  • 𝑣 is the magnitude of the velocity of the body
  • 𝑐 is the speed of light in vacuum
  • 𝑚0 is the rest mass of the body.

The value 𝑚 is known as the relativistic mass of the body. The factor $\dfrac 1 {\sqrt{1 - \dfrac {v^2} {c^2} } }$ is known as the Lorentz Factor.

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:

$$\frac{\mathrm d}{\mathrm d t} \left({\frac {\mathbf v}{\sqrt{1 - \dfrac {v^2}{c^2} } } }\right)$$ $$ = \frac{\mathrm d}{\mathrm d v} \left({\frac {\mathbf v}{\sqrt{1 - \dfrac {v^2}{c^2} } } }\right) \frac{\mathrm d v}{\mathrm d t} $$ $$ = \mathbf a \left({\frac {\sqrt{1 - \dfrac {v^2}{c^2} } - \dfrac v 2 \dfrac 1 {\sqrt{1 - \dfrac {v^2}{c^2} } } \dfrac{-2 v}{c^2} } {1 - \dfrac {v^2}{c^2} } }\right) $$ $$ = \mathbf a \left({\frac {c^2 \left({1 - \dfrac {v^2}{c^2} }\right) + v^2} {c^2 \left({1 - \dfrac {v^2}{c^2} }\right)^{3/2} } }\right) $$ $$ = \mathbf a \left({\frac 1 {\left({1 - \dfrac {v^2}{c^2} }\right)^{3/2} } }\right)$$

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

Sources

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