Projectile Motion Calculator

Calculate the trajectory, range, maximum height, and flight time of a projectile launched at an angle.

Projectile Motion Formulas: Horizontal Motion: • x = v₀cosθ × t • vₓ = v₀cosθ (constant) Vertical Motion: • y = h₀ + v₀sinθ × t - ½gt² • vᵧ = v₀sinθ - gt Key Results: • Maximum Height: H = h₀ + (v₀sinθ)² / (2g) • Flight Time: t = [v₀sinθ + √((v₀sinθ)² + 2gh₀)] / g • Range: R = v₀cosθ × t • For level ground (h₀=0): R = v₀²sin(2θ) / g Where: • v₀ = initial velocity • θ = launch angle • h₀ = initial height • g = gravitational acceleration (9.81 m/s²) • t = time
Example 1 (45° optimal angle): v₀ = 20 m/s, θ = 45°, h₀ = 0 m Range = 20² × sin(90°) / 9.81 = 40.8 m Max Height = 400 × 0.5 / 19.62 = 10.2 m Flight Time = 2 × 20 × 0.707 / 9.81 = 2.88 s Example 2 (Basketball shot): v₀ = 8 m/s, θ = 50°, h₀ = 2 m (release height) Horizontal v = 8 × 0.643 = 5.14 m/s Vertical v = 8 × 0.766 = 6.13 m/s Max Height ≈ 3.9 m Range ≈ 6.3 m Example 3 (Soccer kick): v₀ = 25 m/s, θ = 30°, h₀ = 0 m Range = 625 × sin(60°) / 9.81 = 55.1 m Max Height = 625 × 0.25 / 19.62 = 8.0 m Flight Time = 2.55 s Example 4 (Cliff launch): v₀ = 15 m/s, θ = 40°, h₀ = 50 m Horizontal v = 11.49 m/s Initial vertical v = 9.64 m/s Max Height ≈ 54.7 m Range ≈ 42 m (extended by height) Flight Time ≈ 3.7 s Example 5 (Water fountain): v₀ = 10 m/s, θ = 60°, h₀ = 0.5 m Max Height ≈ 4.3 m Range ≈ 9.2 m Flight Time ≈ 1.82 s

What is projectile motion?

Projectile motion is the motion of an object thrown or projected into the air, subject only to gravity. It follows a parabolic path with independent horizontal and vertical motion components.

What assumptions does this calculator make?

This calculator assumes: no air resistance, constant gravitational acceleration (9.81 m/s²), flat ground (unless initial height specified), and projectile is a point mass. Real projectiles experience air drag and wind.

What is the optimal angle for maximum range?

On level ground (zero initial height), 45° gives maximum range. With initial height, slightly less than 45° is optimal. Into wind or uphill, use lower angles; downhill or with tailwind, higher angles may work better.

How do I calculate maximum height?

Maximum height H = h₀ + (v₀sinθ)² / (2g), where h₀ is initial height, v₀ is initial velocity, θ is launch angle, and g = 9.81 m/s². Height is reached at half the flight time.

How do I calculate range?

Range R = v₀cosθ × t_total, where t_total is total flight time. Alternatively, R = v₀²sin(2θ)/g + horizontal distance from height difference. Horizontal velocity is constant (no air resistance).

What is flight time?

Flight time is how long the projectile is airborne. It depends on vertical motion only: t = [v₀sinθ + √((v₀sinθ)² + 2gh₀)] / g. Time up + time down = total time.

Why doesn't air resistance matter for some projectiles?

Air resistance is negligible for dense, slow-moving objects over short distances (e.g., shot put). For light, fast, or long-distance projectiles (bullets, golf balls), air resistance significantly affects trajectory.

How does initial height affect range?

Greater initial height increases range and flight time. A projectile launched from height h₀ has more time to travel horizontally before hitting the ground, extending its range beyond the level-ground case.

Can I use this for sports analysis?

Yes, for approximations in basketball, soccer, baseball, golf, javelin, shot put, etc. Real sports involve spin, air resistance, and surface conditions, but this gives good initial estimates for trajectory planning.

What are real-world examples of projectile motion?

Basketball shots, football passes, artillery shells, water fountains, fireworks, long jump, ski jumping, cliff diving, and object drops from aircraft. Any object moving under gravity alone follows projectile motion.

How do I convert between units?

Velocity: 1 m/s = 3.6 km/h = 2.237 mph. Distance: 1 m = 3.281 ft. Common speeds: baseball pitch ~40 m/s, soccer kick ~30 m/s, basketball shot ~8 m/s.

What happens at 90° (straight up)?

At 90°, horizontal velocity is zero, so range is zero. The projectile goes straight up and comes straight down. Maximum height is h₀ + v₀²/(2g), and time aloft is 2v₀/g.