A projectile launched with speed v₀ at an angle θ separates into two independent motions. Horizontally it moves at a constant v₀·cos(θ); vertically it accelerates downward at g, starting at v₀·sin(θ). Gravity bends the path into a parabola, and the same launch can produce very different flights depending on angle and gravity.
Separating projectile motion into horizontal and vertical partsRange R = v₀² · sin(2θ) / g
Max height H = (v₀·sinθ)² / (2g)
Time of flight T = 2·v₀·sinθ / g
The 45° rule
For a fixed speed on level ground, the range is greatest when θ = 45°. Lower angles fly flatter but shorter; steeper angles climb higher but land sooner.