Launch at an angle to the horizon
The motion of a body thrown with speed v₀ at an angle α to the horizon splits into two independent ones: uniform along the horizontal with the speed v₀x = v₀ cos α and uniformly accelerated along the vertical with the acceleration −g and the initial speed v₀y = v₀ sin α. The coordinates are x = v₀x t and y = h₀ + v₀y t − g t² / 2. Eliminating the time gives the trajectory, a parabola y = h₀ + x tan α − g x² / (2 v₀² cos²α).
Equations for a launch from the ground (h₀ = 0)
| Quantity | Formula |
|---|---|
| Time up | t₁ = v₀ sin α / g |
| Time of flight | t = 2 v₀ sin α / g (time up equals time down) |
| Maximum height | H = v₀² sin²α / (2g) |
| Range | L = v₀² sin 2α / g |
| Speed at impact | equal to the initial one, the impact angle equals the launch angle |
The range is greatest at α = 45°: L = v₀²/g. The angles α and 90° − α give the same range but different height and time (flat and lofted trajectories).
Horizontal launch
For α = 0 from the height h the time of fall t = √(2h / g) does not depend on v₀, the range is L = v₀ t and the impact speed is v = √(v₀² + 2gh). From 20 m with g = 10 and v₀ = 15 m/s: t = 2 s, L = 30 m, v = 25 m/s.
Launch from a height
For h₀ > 0 the time of flight is the positive root of h₀ + v₀y t − g t²/2 = 0. By energy conservation the impact speed is v = √(v₀² + 2g h₀) whatever the angle. The greatest range is at an angle below 45°: tan α = v₀ / √(v₀² + 2g h₀).
Air resistance
In air the drag of a ball is F = ½ ρ C S v², the deceleration is k v², k = ρ C S / (2m). There is no closed formula, the path is computed numerically. The result: a shorter range and height, a rise shorter than the fall, a steeper fall, and the best angle below 45°. The effect is large for a light ball and small for a heavy shot. The calculator integrates the flight of a ball by the Runge-Kutta method and compares it with the flight in vacuum.