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I have a body that is being affected by forces, some of which are invariant ($F_i$) and some of which vary with velocity ($F_v(v)$). Given that $F=ma$ (where $F=F_i+F_v(v)$), how do I integrate this to find a total displacement once the body accelerates to a speed $v$?

I'm using Wolfram Cloud to try to solve this problem. I tried to use the DSolve function to do this:

DSolve[s''[v]=F/m, s[v], v]

but it's not giving me sane answers. I think that's probably because it has no idea that v is s', but I'm not sure if this is the right way to go about it.

I'd appreciate any help in understanding how to turn this PDE into an equation for total distance traveled at the point a velocity is reached.

EDIT: Here's the actual equation describing an airplane's takeoff...

$F$ combines the following:

  • Thrust generated by the propeller producing $P$ thrust horsepower ($F=\frac{P}{v}$)
  • Rolling resistance generated by the wheels ($D_r=c W_w$ where $W_w$ is the weight on wheels [$W_w=W-\frac{1}{2} A_w C_L \rho v^2$])
  • Parasite drag generated by the airflow ($D_p=\frac{1}{2} A_c C_D \rho v^2$)

Thus total thrust is dependent on velocity.

In Wolfram Cloud, I have combined these into a term F, and then I solve like so:

DSolve[{(F/.{v->v[t]})==m*v'[t], v[0]==0},v[t],t]

(the ReplaceAll operation /. is used to convert the variable v into a functional variable v[t])

This in turn is producing an unusable, ungainly result.

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I don't really see the problem, why don't you just solve your differential equation? Maybe more info would be helpful.

If you're having trouble, perhaps just calculate your $v(t)$:

$F = F_0 + F_v (v) = m\dot{v}$ (solving this you can calculate the time needed to reach a certain velocity)

Taking the derivative of your $v(t)$ you should be able to solve for your $x(t)$.

EDIT: Also not sure what you mean by partial DE, as far as I see $a$ and $v$ are both time derivatives.

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  • $\begingroup$ I've updated the original question with details specific to my problem. I tried what you said, applying DSolve in terms of v[t] rather than s''[v], but unfortunately the solution is so complex that Wolfram One is not calculating it. I may have to simplify my terms. $\endgroup$ – RISCfuture Aug 9 at 19:45
  • $\begingroup$ P.S. If it's not obvious from my original wording, I have a poor and fragmented memory of DiffEq from college... >< $\endgroup$ – RISCfuture Aug 9 at 19:46
  • $\begingroup$ So a quadratic law of resistance and a constant power of thrust, I've had a similar problem in my second year. Basically you get something like P/v + Cv² = m dv/dt . As far as I remember I tried iterating it, but I believe you can get away with solving it analytically for the most part (helps if you put it in a dimension-less form form). I'll scribble a bit and get back to you if I find something that could help you. Good luck! $\endgroup$ – Matej Bajec Aug 9 at 20:29

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