🏃 Projectile Motion — Optimal Launch Angle

Physics of Projectile Motion

Ignoring air resistance, horizontal and vertical motion are independent. The equations are:
x = v₀ cosθ · t  |  y = v₀ sinθ · t − ½ g t²

Setting y = 0 gives the time of flight T = 2 v₀ sinθ / g, and the range R = v₀² sin 2θ / g. Since sin 2θ is maximised at 2θ = 90°, the optimal angle is 45° for level ground with no air resistance.

With air resistance the drag force Fₐ = ½ ρ Cₐ A v² (where A = πr² is cross-sectional area) slows the ball and shifts the optimal angle below 45° — typically to 30°–40° depending on speed and ball properties.

The range envelope curve (dashed) connects the tips of all possible trajectories at a given launch speed — every point inside it can be hit at two different angles; points outside it are unreachable. The envelope itself is an ellipse with the equation R·g = v₀² − (gx)²/v₀² (parabola of safety).

Magnus effect: Spinning balls (footballs, golf balls, baseballs) experience an additional lift/deflection force perpendicular to their velocity, curving trajectories away from the simple parabola seen here.

Sports applications

A shot-put athlete should aim close to 42° (drag reduces the optimum). A football kicked at 30° travels farther than one kicked at 60° — both have the same vacuum range, but air resistance punishes higher, slower arcs more. In archery and javelin, competitors intuitively converge on these drag-adjusted optima.