# Projectile Velocity with Drag

I'm trying to determine the velocity profile (and trajectory) of a small projectile that is fired with an initial velocity. I'm implementing it in Mablab and my equation isn't working out and I just can't figure out what the problem is. The reduction in velocity I get in the first step is greater than the initial velocity.

$$ma = \frac{-CAdv^2}{2}$$

where C is the drag coefficient, A is the projected area, d is the air density, and v is the velocity. Then,

$$\frac{dv}{dt} = \frac{-CAdv^2}{2m}$$ $$v(i)-v(i-1) = \frac{-CAdv^2}{2m}dt$$ $$v(i) = v(i-1) - \frac{CAdv^2}{2m}dt$$

My initial velocity is 1000 m/s and these are 12 mm steel cubes. This starts supersonic and drops through transonic and subsonic regimes, but I've found that C of 0.3 is an ok drag coefficient to experiment with. A is 1.44 (10^-4) m^2, d = 1.225 kg/m^3, m = 1.4 (10^-2) kg. I'm just leaving dt at 1 s for now.

Using this approach results in my drag loss being greater than my initial velocity (I lose -1914 m/s to drag in the first step! That's a lot considering my starting velocity is 1000 m/s. Can anyone please tell me what I'm missing here? I think I've looked at it so much that I'm only capable of looking at it the same way.

Hint : $$dv/dt=-v^2$$ can be reformulated as $$dv/v^2 = -dt$$ Then integrate each side etc.