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Hi all,
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.
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.