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Three forces are acting on an object.

Gravity, mg

Lorentz force, av(a is a constant)

Air resistance, kv^2( k is a constant)

v is the velocity of the object

I figured that this differential equation must be the right one:

M(total)[itex]\frac{dv}{dt}[/itex]= mg-av-kv^2

Where m and M(total) are different masses

[itex]\frac{dv}{kv^2+av-mg}[/itex] = -[itex]\frac{dt}{M(total)}[/itex]

Integrate both sides:

-[itex]\frac{2}{\sqrt{a^2+4kmg}}[/itex] artanh([itex]\frac{2kv+a}{\sqrt{a^2+4kmg}}[/itex]) = -[itex]\frac{T}{M(total)}[/itex] + C

So I want to find the expression for velocity v.

How do I begin? And should the integration constant even be there? When I put limits on the integral(v goes from 0 to v and t goes from 0 to t) then there is no constant of integration.

Thanks so much.

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# Differential equation/rational function

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