Projectile Motion Equations
Projectile motion splits into independent horizontal (constant velocity) and vertical (constant acceleration) components. Gravity acts only vertically at g = 9.81 m/s².
Key Formulas
Components:
v_x = v₀ cos(θ) v_y = v₀ sin(θ)
Time of flight: T = 2v_y / g = 2v₀ sin(θ) / g
Range: R = v_x × T = v₀² sin(2θ) / g
Max height: H = v_y² / (2g) = v₀² sin²(θ) / (2g)Worked Example
Ball: v₀=20 m/s, θ=45°
v_x = 20 cos(45°) = 14.14 m/s
v_y = 20 sin(45°) = 14.14 m/s
T = 2×14.14/9.81 = 2.88 s
R = 14.14 × 2.88 = 40.7 m
H = 14.14²/(2×9.81) = 10.2 mOptimal Angles
Maximum range: θ = 45°
Complementary angles give equal range:
30° and 60° → same R (different T and H)
At 45°: R = v₀²/g
v₀=20 m/s: R = 400/9.81 = 40.8 m ✓Calculate projectile motion: Free Projectile Motion Calculator
Projectile Motion Quick-Reference Table
| Launch angle | Initial speed (m/s) | Max range (m) | Max height (m) | Time of flight (s) |
|---|---|---|---|---|
| 15° | 20 | 20.4 | 1.37 | 1.06 |
| 30° | 20 | 35.3 | 5.10 | 2.04 |
| 45° | 20 | 40.8 | 10.2 | 2.89 |
| 60° | 20 | 35.3 | 15.3 | 3.53 |
| 75° | 20 | 20.4 | 18.8 | 3.95 |
How Projectile Motion Works
Projectile motion treats horizontal and vertical movement independently. Horizontally, there is no acceleration (ignoring air resistance): x = v₀ cos(θ) · t. Vertically, gravity decelerates upward motion and accelerates downward: y = v₀ sin(θ) · t − ½gt². The maximum range occurs at 45° launch angle for flat ground with no air resistance. Complementary angles (e.g., 30° and 60°) produce the same range but different heights and flight times.
Applications include artillery and ballistics, sports biomechanics (optimal javelin and shot-put angles), water fountain design, crash reconstruction, and video game physics engines. In real engineering, air resistance and wind require numerical integration, but the ideal model provides essential first estimates.
Common Mistakes
- Using g = 9.81 vs 10: 9.81 m/s² is the correct standard gravity. Using 10 oversimplifies; answers will be off by ~2%.
- Forgetting to decompose velocity: Always split initial velocity into vₓ = v₀cosθ and v_y = v₀sinθ before applying equations.
- Ignoring launch height: If launched from above ground level, flight time and range both increase. The simple range formula R = v₀²sin(2θ)/g only applies for same-height launch and landing.
Frequently Asked Questions
Range = v₀²sin(2θ)/g. sin(2θ) is maximised when 2θ = 90°, i.e., θ = 45°. This splits the initial speed equally between horizontal and vertical components, achieving the optimal trade-off between launch height (which extends air time) and horizontal speed (which covers distance).
Air resistance reduces both horizontal speed and the effective range. The drag force depends on v², so fast-moving projectiles are slowed disproportionately. For most sport and artillery scenarios with significant drag, the optimal launch angle is less than 45° — typically 30°–40° depending on projectile shape and speed.
Biomechanists analyse high-speed video of javelin throws, long jumps, basketball shots, and soccer kicks to measure launch angle, initial speed, and flight trajectory. This data helps coaches refine technique — a javelin thrower who increases launch speed from 25 m/s to 27 m/s at 35° angle gains several metres of range due to the quadratic relationship between speed and range.