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Projectile motion describes the curved path of an object launched into the air and subject only to gravity (ignoring air resistance). The key insight is that horizontal and vertical motion are completely independent: horizontal velocity is constant (no air resistance, no horizontal force), while vertical motion follows constant gravitational acceleration g = 9.81 m/s² downward.
Given initial speed v₀ and launch angle θ above the horizontal, the three key results are: Range R = v₀²·sin(2θ)/g (horizontal distance traveled), Maximum height H = v₀²·sin²(θ)/(2g), and Time of flight T = 2v₀·sin(θ)/g. These assume the landing point is at the same height as the launch point.
The maximum range for a given initial speed is achieved at θ = 45°, since sin(2θ) = sin(90°) = 1. Complementary angles give the same range: θ = 30° and θ = 60° both produce the same horizontal range (though different flight times and maximum heights). At 90°, the projectile goes straight up and comes straight back down with zero horizontal range.
Worked example: A ball is launched at v₀ = 20 m/s at θ = 30°. Range R = 20²×sin(60°)/9.81 = 400×0.866/9.81 ≈ 35.3 m. Max height H = 400×sin²(30°)/(2×9.81) = 400×0.25/19.62 ≈ 5.1 m. Time of flight T = 2×20×sin(30°)/9.81 = 20/9.81 ≈ 2.04 s.
R = v₀²·sin(2θ)/g | H = v₀²·sin²(θ)/(2g) | T = 2v₀·sin(θ)/g
Enter initial speed, launch angle, and optional initial height to compute all projectile quantities.
Projectile Motion Formulas
| Quantity | Formula (h₀ = 0) | Notes |
|---|---|---|
| Horizontal velocity | vx = v₀cos(θ) | Constant throughout |
| Vertical velocity | vy = v₀sin(θ) − gt | Decreases due to gravity |
| Max height | H = v₀²sin²(θ)/(2g) | When vy = 0 |
| Time of flight | T = 2v₀sin(θ)/g | Total air time (h₀=0) |
| Horizontal range | R = v₀²sin(2θ)/g | Maximum at θ = 45° |
| Optimal angle | θ = 45° | Maximises range on flat ground |
This calculator is for educational purposes only and does not constitute professional advice. Results are based on standard mathematical formulas. Always verify critical calculations with a qualified professional before making important decisions.