Odd Force Equation

[foxgloves]

I was reading a book, and I ran into this odd equation:

F=\frac{d}{dt}(\frac{mv}{(1-\frac{v^2}{c^2})})

Where does this equation come from and what does it represent?

[bosonicyouth]

It's the relativistic version of Newton's second law. If the velocity v << c, where c is the speed of light, the denominator vanishes and you obtain the Newtonian version F=ma. However, as an object approaches the speed of light the denominator approaches zero, and the force required to further speed it up approaches infinity. So this always enforces the universal speed limit of c for massive objects and is consistent with other relativistic effects such as time dilation and length contraction, all of which contain the Lorentz factor in some form, as seen here also (only squared).

[Qaanol]

Yeah, the “real” version of Newton’s 2nd law, as in the version that’s true and generally applicable, is “Force equal rate of change of momentum”, or symbolically F=dp/dt. That is the definition of “force”. You’ll never see a proof of Newton’s 2nd law, because it is not a theorem but actually a statement of definition.

So, once you know that force is defined as F=dp/dt, then calculating force just means taking the derivative of momentum. And as it turns out, in special relativity the momentum of an object is p=γmv, where γ is the Lorentz factor γ=dt/dτ=1/√(1-v²/c²).

[eSOANEM]

The thing is, the lorentz factor has a square root in it so what foxgloves has actually written is very odd (and will only hold at rest). I guess they made typo/transcribed it incorrectly.