Launch a projectile and explore range, trajectory, and maximum height. Compare ideal parabolic motion with realistic air resistance. Discover why 45° maximises range in vacuum but not in air.
Without drag: x = v₀cosθ·t, y = v₀sinθ·t − ½gt². Range R = v₀²sin(2θ)/g, maximised at θ = 45°. With drag: F_d = −½ρv²C_d·A, giving acceleration a = g + F_d/m. Numerical RK4 integration handles the drag term. With air resistance, the optimal angle drops below 45° (≈38° for dense objects in air).
Drag force is quadratic in velocity: F_d ∝ v². This means fast-moving objects experience much stronger drag than slow ones. A cannonball and a tennis ball launched at the same angle and speed will land very differently — the tennis ball experiences far more drag relative to its weight (lower ballistic coefficient m/C_d·A).
In vacuum, 45° always maximises range. In air, the optimum shifts toward lower angles (35–42°) because the longer arc at higher angles spends more time fighting drag. Shot put athletes use ≈42° for this reason. Arrow archers often use flatter angles (20–30°) because arrows have extremely low drag.