Since I’m horrible at math, but want to improve, I’ve been playing games. My favorite one is a pencil-and-paper version of Scorched Earth–I put in the variables and find out where a projectile, when fired at a certain speed and angle, will land. It’s complicated, having dozens of variables, and a good way to goof off in class, while looking like I’m doing something productive.
I want to add another variable. I’ve tried to add wind resistance, but I, for the life of me, can’t figure it out. I know wind resistance works exponentially–if the projectile went twice as fast, it’d experience four times the drag–but I can’t figure out how many meters per second would be bled from the projectile’s velocity.
I’m assuming the projectile is a perfect sphere, with no protrusions. How do I, given the frontal area of the projectile and the atmospheric viscosity, calculate how many meters per second are bled away?
In the simplest form, air drag is simply D=CdAr*v[sup]/2 where Cd is the coefficient of drag, A is the frontal cross-section area, r is the air density and v is velocity. Cd is usually a constant that depends on the shape of the object, but it also changes slightly due to other conditions. For example, Cd for supersonic speed will not be the same as the Cd for subsonic speed even for the same object. Note that Cd is a dimensionless coefficient. Just stay within the same unit system (e.g. if you choose MKS, use meter, kg, second, Newtons, etc) and you’ll be fine.
Once you know the force, loss of speed (rate of deceleration) is calculated from Newton’s Laws (v’ = D / M, where v’ is the derivative of velocity, i.e. acceleration, D is drag force from the first equation, and M is mass.)
You might want to see what some high school physics books say (at the library, a local high school, etc.). I once saw a polynomial formula (ie: v^2 +v) and the wind resistance could be taken as a linear function at very low speeds. We didn’t spend much time on this, and I think it was a quick and dirty way to approximate terminal velocities (and the wind resistances you may be seeking.)
I just thought I’d offer this as food for thought. I took the course about 20 years ago! (I still use the physics, but not that specific formula, so physics class remains fresh in my memory.)