## So I have gotten this far... $a = \left(\left({F}_{a} - \frac{{c}_{d} \cdot {\rho}_{h} \cdot A \cdot {v}^{2}}{2}\right) - {m}_{t} \cdot {g}_{r}\right)$ This is based upon ${F}_{a} - \frac{{F}_{d} + {F}_{g}}{m} = a$ But $a$ is $\frac{\mathrm{dv}}{\mathrm{dt}}$ So this becomes a differential equation because the faster you go the harder it is to go faster. How can one solve this? Please note I'm currently a Calc AB student.

Jul 23, 2016

see below

#### Explanation:

your work looks a little bit messed up. EG It is dimensionally incorrect for starters.

I can see the sense of it, the interaction between Rayleigh drag and gravity. Is there also an additional force that is purporting to act on the mass? i can't figure that out on a quick look.

the DE you get shouldn't be too much trouble, in the sense it should be separable. the problem you will have is the same as with all damping/ friction like effect in that it opposes motion and so you becomes obsessed with the actual sign of v

and $F = m g - \frac{1}{2} \setminus \rho {v}^{2} \setminus {C}_{D} \setminus A = m \frac{\mathrm{dv}}{\mathrm{dt}}$