Projecta

Meet Projecta, a projectile who flies farthest when launched at 45°.

With no air resistance, a thrown object moves sideways at constant speed while gravity accelerates it downward. The two motions are independent. Combining them gives a parabolic path, and on level ground the range is R = v² sin 2θ / g.

The sin 2θ factor peaks at 1 when θ = 45°, so that angle goes farthest. Angles equally above and below 45° give the same range: 30° and 60° land at the same spot, both at sin 60° ≈ 86.6 percent of the 45° range. Launched at 20 m/s and 45°, a ball would travel about 20² / 9.8 ≈ 41 metres.

Engineers and sports scientists use the same model as a starting point for anything thrown, kicked or fired.

Air resistance changes the answer. With drag, the best angle for a long throw is usually below 45°, and real balls also spin. Launching from a height, as a shot-putter does, also lowers the best angle.