I've been attempting to derive the relativistic velocity equation (for an object accelerating with a constant force- [tex]\frac{Fct}{\sqrt{1-\frac{v^{2}}{c^{2}}}}[/tex]) for near a month without any solution.(adsbygoogle = window.adsbygoogle || []).push({});

I've derived an equation form the F=ma relation which includes a Sin function, so I know it's wrong. However, I haven't been able to determine why. Could someone point me in the right direction?

My derivation:

Using [tex]a=\frac{F}{m}[/tex] and [tex]a\equiv\frac{dv}{dt}[/tex]

[tex]\frac{dv}{dt}=\frac{F}{m}[/tex]

[tex]m dv= F dt[/tex] where [tex]m\equiv\frac{m_{o}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}[/tex]

[tex]\frac{m_{o} dv}{\sqrt{1-\frac{v^{2}}{c^{2}}}}= F dt[/tex]

Integrate both sides

[tex]m_{o}c Sin^{-1}(\frac{v}{c})=Ft + constant[/tex]

Which gives an answer that doesn't make sense and obviously doesn't represent v(t)

[tex]\frac{v}{c}=Sin(\frac{Ft + const}{m_{o}c})[/tex]

Any idea where the error is, or how to get the correct expression? Any help is appreciated.

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# Relativistic velocity derivation under constant force

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