Acceleration Model

I also posted this in one of the engineering forums, but it is essentially a math problem, so I thought I would post it here as well.

I am working on a model of a solar powered drag race.

The race: 250 meters, no incline, initial velocity = 0.

Classical physics gives us the following equations (ignores aero and rolling drag and wheel rotation):

Velocity as a function of time V = (2Pt/M)^.5
P = power, M = mass

Distance as a function of time d = (2/3)((2P/M)^.5)t^(1.5)

Time to travel x distance t = ((1.5d)^(2/3))(M/2P)^1/3


But of course we do lose power to aerodynamic drag forces

Pa = .5rCdAV^3 (r = air density, Cd = aero drag coef., A = frontal area)

and rolling drag at the wheels

Pr = CrMV (Cr = rolling drag coef.)

and wheel rotation

Pw = FwV^3 (Fw = wheel rotational factor)


So if our inertial power equals our power in, Pi (from the solar panel) minus our power lost to friction (aero, rolling, wheel rotation), then our velocity equation becomes:

V = (2((Pi-(.5rCdAV^3)-(CrMV)-(FwV^3))t)/M)^.5

I need help solving this equation. I have approximated the solution by chopping the race up into small pieces and solving iteratively, but I would rather do it right.
 

fresh_42

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Let me sort this out.
Our variable is the velocity function in time: ##v(t)##.
We have a kinetic energy ##E## which you call initial power Pi.
We have air resistance ##P_a = \alpha \cdot v(t)^3## with some constant factor ##\alpha## which you wrote rCdA.
We have a drag of ##P_r= \beta \cdot v(t)## with some constant factor ##\beta## which you wrote CrM.
We have rolling friction ##P_w= \gamma v(t)^3## with some constant factor ##\gamma## which you wrote Fw.

Given your formulas are correct, we end up with ##(\alpha + \gamma) \cdot v(t)^3 + \frac{M}{2}\cdot v(t)^2 + \beta \cdot v(t) - E = 0## which is a bit tricky to solve, cp. https://en.wikipedia.org/wiki/Cubic_function#Cardano's_method
It probably will be easier to use one of these:
 

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