Projectile motion

A projectile is an object that, once thrown, moves under gravity alone; air resistance is neglected. Its motion is a uniform horizontal motion and a uniformly accelerated vertical motion, independent of each other.

Launched at speed v₀ at angle θ₀ above the horizontal: aₓ = 0, aᵧ = −g; x = (v₀ cos θ₀) t and y = (v₀ sin θ₀) t − ½ g t².

The horizontal velocity v₀ cos θ₀ never changes; the vertical velocity vᵧ = v₀ sin θ₀ − g t falls by g every second and is zero at the top.

The path is y = (tan θ₀) x − g x² / [2 (v₀ cos θ₀)²], a parabola.

Time to the top tₘ = v₀ sin θ₀ / g; time of flight T_f = 2 v₀ sin θ₀ / g; maximum height hₘ = (v₀ sin θ₀)² / 2g; horizontal range R = v₀² sin 2θ₀ / g.

The range is largest at θ₀ = 45°, Rₘ = v₀²/g. Angles 45° + α and 45° − α (complementary angles such as 30° and 60°) give the same range but different heights and times.

A ball thrown at 20 m/s at 30° (g = 9.8 m/s²) rises 5.1 m, stays up 2.04 s and lands 35.3 m away.

Thrown horizontally from a height h, a body falls for t = √(2h/g) whatever its horizontal speed: from 4.9 m it takes 1.0 s, and at 10 m/s it lands 10 m away moving at 14 m/s.