Fetching contributors…
Cannot retrieve contributors at this time
68 lines (54 sloc) 2.47 KB

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

You can’t perform that action at this time.