I've been working on graphing the movements of a sphere when thrown. I'm trying to simulate air resistance. I've found the equation $$F_d=\frac12 C_d\rho V^2A$$
I've separated the velocity into x and y components, and have an equation for vertical displacement from horizontal displacement. I just need to calculate drag force, and that to my equation. I'm stuck on how to calculate velocity. I know that the derivative of position with respect to time is equal to velocity, but I'm trying to calculate position, so I can't use position.
I don't think I can calculate position, as it seems like this only has a numerical solution, so how can I approximate position, in that case.
Is there any way to find velocity as well?
I saw a graph in Desmos of air resistance, so it is possible to calculate it without programming it.
Best Answer
Let's do this with the vertical component. Newton's 2nd law says $F=ma=-mg-\frac{1}{2}C_d\rho Av^2$. For simplicity, let's get rid of all those constants and just call it $a=-1-v^2$. Since $a=dv/dt$, we have $$\frac{dv}{dt}=-1-v^2\\ \frac{dv}{1+v^2}=-dt\\ \text{Tan}^{-1}(v)=-t+c_1\\ v=\frac{dy}{dt}=-\tan(t+c_1)\\ y=-\int tan(t+c_1)dt=\ln|\cos(t+c_1)|+c_2 $$ Now you just plug in initial conditions to find the values of $c_1,c_2$ and you should be set!
For the horizontal case, the stripped-down differential equation should just look like $a=-v^2$. Try solving this on your own!