I am trying to find the effect of air resistance on the projectile motion of an ice skater performing a jump (I am aware it technically isn't called a projectile motion once there is air resistance involved but for the sake of explaining it shortly, I will say it is).
I have found that $F\text{net} = m*a = kv^2 - mg$ , since gravity is acting downwards and air resistance acting upwards, k being some constant for air resistance. So for the math part:
$$\frac{dv} {((kv^2)/m - g)} = dt$$
I have done integration by substitution at first, by using u = kv^2/m - g and $du = (\frac{2kv}m)dv$ , however, I am not sure if this is the correct approach and even if it is, I am unable to solve the final equation I found for $v$:
$$\ln\left(\frac{kv^2}m - g\right)= \frac{t2kv}m$$
steps I followed to integrate and solve for v
I essentially followed the steps I outlined in the attached image
If anyone can help I would appreciate it so much. Thank you!
Simply use this Riccati's equation as a special type of first-order nonlinear differential equation and the solution $$ v(t)=-\frac{\sqrt{b} \tanh\big(\sqrt{ab} (c_1+t)\big)}{\sqrt{a}} $$ where
Mind that arctanh has a special relationship to the logarithm.
Some straightforward rewrite and the results are as expected. $arctanh$ is the inverse of $tanh$.
Somewhat comfortable is this with g, k and m.
To do the solution is documented in done by wolframaplha too but in the paid for version.
There are of course sides that show the solution. For example ode0123. That is a specialized side for ordinary differential equations.
This page offers some of the newest tricks available for Ricatti: Riccati Differential Equation.