Side view of a snowball with quadratic air drag, solved with RK4 and compared with the vacuum parabola.
The science
Gravity gives a constant downward acceleration. Air resistance on a sphere is a quadratic drag force F = 1/2 rho Cd A v^2 opposite to the velocity relative to the air (rho = 1.225 kg/m3, Cd = 0.47 held constant). Per unit mass this is k |v_rel| v_rel with k = rho Cd A / (2 m), and the equations are advanced with a fourth-order Runge-Kutta scheme (dt = 2 ms).
Without drag the range from the ground is v^2 sin(2a) / g, best at 45 degrees. Drag shortens the flight, lowers the best angle and makes the descent steeper than the climb. A falling snowball approaches the terminal speed sqrt(2 m g / (rho Cd A)), where drag balances weight; light powder snow has a low terminal speed, an icy ball a high one.
Simplifications: constant Cd (real Cd changes with Reynolds number), no spin or lift, no snow crumbling in flight, uniform wind. The two numerical checks use the vacuum formula and the terminal speed.