Projectile Range Calculator
Calculate the horizontal range, maximum height and time of flight of a projectile launched from flat ground using R = v₀²sin(2θ)/g. Distinct from the trajectory plotter — this gives numerical answers only.
How to use this tool
- Enter launch velocity v₀, launch angle θ and gravity g in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your horizontal range and the full breakdown beneath it.
For a projectile launched from level ground: R = v₀²sin(2θ)/g. Maximum range occurs at θ = 45°. Max height: H = v₀²sin²θ/(2g). Time of flight: T = 2v₀sinθ/g. Ignores air resistance.
Formula
Horizontal range: R = v02 × sin(2θ) / g
Maximum height: H = v02 × sin2(θ) / (2g)
Time of flight: T = 2 v0 sin(θ) / g
How it works
This calculator uses the standard kinematic equations for a projectile launched from flat ground with initial speed v0 at angle θ above horizontal, under constant gravitational acceleration g. Air resistance is ignored, so results represent the ideal vacuum trajectory. The maximum possible range at a given speed occurs at exactly 45° and equals v02/g.
Worked example
Worked example
- Inputs: v₀ = 20 m/s, θ = 45°, g = 9.81 m/s². Convert angle to radians: 45 × π/180 = 0.7854 rad.
- Range: R = 20² × sin(90°) / 9.81 = 400 × 1 / 9.81 = 40.775 m.
- Max height: H = 400 × sin²(45°) / (2 × 9.81) = 400 × 0.5 / 19.62 = 10.194 m.
- Time of flight: T = 2 × 20 × sin(45°) / 9.81 = 40 × 0.7071 / 9.81 = 2.884 s.
Horizontal range = 40.775 m; maximum height = 10.194 m; time of flight = 2.884 s.
Key terms
- Launch velocity (v₀)
- The speed of the projectile at the moment of launch, before gravity acts on it.
- Launch angle (θ)
- The angle above the horizontal at which the projectile is fired. 45° maximises range on flat ground.
- Time of flight
- The total time the projectile is in the air from launch until it returns to the same height.
- Maximum height
- The peak vertical position reached during the trajectory, where vertical velocity is momentarily zero.
- Gravitational acceleration (g)
- The downward acceleration due to gravity; approximately 9.81 m/s² on Earth's surface.
Frequently asked questions
- How does this differ from the projectile motion calculator?
- That tool plots the trajectory chart and supports non-zero launch height. This calculator focuses on exact numerical outputs from the standard range formula.
- What launch angle gives the greatest range?
- 45° on flat ground with no air resistance. Air resistance shifts the optimal angle slightly below 45°.