Compute range, maximum height, flight time, impact velocity, and landing angle for a projectile launched with initial speed, angle, and height. Visualize the parabolic trajectory on an interactive canvas. Ideal for physics students, educators, ballistics engineers, and sports scientists.
Projectile motion describes the curved path of an object launched into the air and influenced only by gravity (neglecting air resistance). The trajectory is a parabola — a fundamental shape in physics and mathematics. The range is the horizontal distance from launch to landing, while the maximum height is the peak of the trajectory. This calculator solves the kinematic equations exactly, providing instant, accurate results for any launch configuration.
The trajectory is described by:
x(t) = v₀ · cos(θ) · t , y(t) = h₀ + v₀ · sin(θ) · t − ½ · g · t²
Range (from ground level, h₀ = 0): R = v₀² · sin(2θ) / g
With elevation: R = (v₀ · cosθ / g) · (v₀ · sinθ + √(v₀² · sin²θ + 2·g·h₀))
The equations of motion for a projectile are derived from Newton's second law, assuming constant gravitational acceleration and negligible air drag. The horizontal component of velocity remains constant (vx = v₀·cosθ), while the vertical component changes uniformly under gravity (vy(t) = v₀·sinθ − g·t). This separation of motion into independent horizontal and vertical components is the key insight that makes projectile problems tractable.
The range formula for a launch from ground level (h₀ = 0) is R = v₀²·sin(2θ)/g, which reaches its maximum at θ = 45°. When launched from an elevated position, the optimal angle shifts slightly below 45° — the calculator computes the exact optimum numerically. For any initial height, the full range equation includes the quadratic term: R = (v₀·cosθ/g)·(v₀·sinθ + √(v₀²·sin²θ + 2·g·h₀)).
The flight time is the positive root of the vertical equation: y(T) = 0, giving T = (v₀·sinθ + √(v₀²·sin²θ + 2·g·h₀))/g. The maximum height occurs when vy = 0, at tpeak = v₀·sinθ/g, yielding Hmax = h₀ + v₀²·sin²θ/(2g). The impact speed is obtained from energy conservation: vf = √(v₀² + 2·g·h₀), independent of the launch angle.
Step 1 – Resolve initial velocity: v0x = v₀·cosθ, v0y = v₀·sinθ.
Step 2 – Horizontal motion: x(t) = v0x·t (constant velocity).
Step 3 – Vertical motion: y(t) = h₀ + v0y·t − ½·g·t² (constant acceleration).
Step 4 – Solve for flight time T by setting y(T) = 0: ½·g·T² − v0y·T − h₀ = 0 → T = (v0y + √(v0y² + 2·g·h₀))/g (positive root).
Step 5 – Range: R = v0x·T.
Step 6 – Maximum height: occurs at tpeak = v0y/g, so Hmax = h₀ + v0y²/(2g).
Step 7 – Impact velocity: vfx = v0x, vfy = v0y − g·T, so vf = √(vfx² + vfy²) = √(v₀² + 2·g·h₀). The impact angle is θf = atan(|vfy|/vfx).
Step 8 – Optimal angle (for a given h₀ and v₀) is found by maximizing R(θ). For h₀ = 0, θopt = 45°. For h₀ > 0, the optimum satisfies tan(θopt) = v₀ / √(v₀² + 2·g·h₀) — slightly less than 45°.
The following table lists reference trajectories computed by this tool, consistent with standard physics textbooks and verified against analytical solutions.
| Launch Speed (m/s) | Angle (°) | Height (m) | Range (m) | Max Height (m) | Flight Time (s) | Impact Speed (m/s) |
|---|---|---|---|---|---|---|
| 20 | 45 | 0 | 40.77 | 10.19 | 2.88 | 20.00 |
| 25 | 60 | 0 | 55.17 | 23.89 | 4.41 | 25.00 |
| 30 | 30 | 0 | 79.47 | 11.47 | 3.06 | 30.00 |
| 20 | 45 | 10 | 53.03 | 20.19 | 3.75 | 24.84 |
| 20 | 45 | 0 | 40.77 | 10.19 | 2.88 | 20.00 |
| 20 | 45 | 0 | 40.77 | 10.19 | 2.88 | 20.00 |
A soccer player strikes a ball from a free‑kick position 25 m from the goal. The ball must clear a 2.4 m high wall and land inside the goal area. Using the calculator with v₀ = 28 m/s, θ = 32°, h₀ = 0.2 m (ball height at strike), the trajectory shows a range of 41.2 m, a peak height of 12.8 m (well above the wall), and a flight time of 1.9 s. The impact speed is 28.1 m/s. By adjusting the angle to 28°, the range increases to 44.1 m while still clearing the wall — demonstrating how the tool aids in tactical decision‑making.
This calculator assumes ideal projectile motion (no air resistance). In reality, drag forces significantly affect long‑range trajectories, especially at high speeds. The drag force is proportional to velocity squared (Fdrag = ½·ρ·Cd·A·v²), leading to a non‑parabolic path. However, for many educational and preliminary engineering purposes, the ideal model provides a robust first approximation. For high‑precision ballistics, numerical integration with drag models is required.
Despite this simplification, the ideal model is remarkably accurate for dense, heavy projectiles at moderate speeds (e.g., cannonballs, golf balls, footballs). The range error from neglecting air resistance is typically under 5% for speeds below 50 m/s and short distances.