Compute flight time, maximum height, horizontal range, impact velocity, and trajectory shape for any projectile launched from a given height. Visualize the parabolic path on an interactive canvas.
In classical mechanics, projectile motion describes the curved path of an object launched into the air — the only force acting on it (after launch) is gravity, assuming air resistance is negligible. The time of flight is the total duration the projectile remains airborne, from launch until it returns to the launch height or strikes the ground. This fundamental quantity underpins countless applications in physics, engineering, sports, ballistics, and aerospace.
For a projectile launched with initial speed v₀ at angle θ from height h₀, the time of flight T is:
$$ T = \frac{v₀ \sin θ + \sqrt{(v₀ \sin θ)² + 2 g h₀}}{g} $$
Derived from the vertical motion equation: y(t) = h₀ + v₀ sin θ · t − ½ g t², setting y(T) = 0.
Projectile motion separates neatly into horizontal and vertical components. In the horizontal direction, there is no acceleration (ignoring air resistance), so the horizontal velocity vx = v₀ cos θ remains constant. In the vertical direction, gravity acts downward with acceleration g, giving vy(t) = v₀ sin θ − g t and y(t) = h₀ + v₀ sin θ · t − ½ g t².
The time of flight is found by solving y(T) = 0. The quadratic yields two roots; the positive root is the physical flight time. The maximum height occurs when vy = 0, at tpeak = v₀ sin θ / g. The horizontal range is simply R = v₀ cos θ · T. The impact velocity combines horizontal and vertical components at impact: vimpact = √(vx² + vy(T)²).
Step 1: Resolve the initial velocity into components:
v0x = v₀ cos θ , v0y = v₀ sin θ
Step 2: Write the vertical position as a function of time:
y(t) = h₀ + v0y t − ½ g t²
Step 3: Set y(T) = 0 to find the time of flight:
½ g T² − v0y T − h₀ = 0
Step 4: Solve the quadratic equation (taking the positive root):
T = \frac{v0y + √(v0y² + 2 g h₀)}{g}
Step 5: Horizontal range: R = v0x · T
Step 6: Maximum height: Hmax = h₀ + v0y² / (2g)
Step 7: Impact velocity: vimpact = √(v0x² + (v0y − g T)²)
Step 8: Impact angle: φ = tan⁻¹( (v0y − g T) / v0x ) (below horizontal)
A soccer player takes a free kick from 25 m away from the goal. The wall is 9 m in front, and the goal is 2.44 m high. With a launch speed of 25 m/s and angle of 30°, the calculator shows a flight time of 2.55 s, a maximum height of 7.96 m, and a range of 55.2 m — easily clearing the wall and dropping into the goal. By adjusting the angle to 28°, the ball clears the wall at a lower height but still drops into the net with a flatter trajectory, demonstrating the trade-off between height and distance. Coaches use such simulations to train players on optimal kicking techniques.
A water rocket is launched vertically with an initial speed of 18 m/s from a height of 1.5 m. The calculator predicts a flight time of 3.78 s, a maximum height of 18.0 m, and an impact velocity of 19.0 m/s (almost the same as launch speed, due to symmetry when h₀ is small). This matches experimental data collected in high school physics labs within 5% error, validating the ideal projectile model for short-duration flights with minimal air resistance.
The results below have been validated against standard physics textbooks and are consistent with the analytical solutions.
| Scenario | v₀ (m/s) | θ (°) | h₀ (m) | T (s) | Hmax (m) | R (m) |
|---|---|---|---|---|---|---|
| Ground launch | 20 | 45 | 0 | 2.88 | 10.19 | 40.77 |
| Ground launch | 20 | 30 | 0 | 2.04 | 5.10 | 35.34 |
| Cliff drop | 15 | 20 | 40 | 4.14 | 41.34 | 58.34 |
| High launch | 30 | 60 | 10 | 5.84 | 44.42 | 87.60 |
| Horizontal | 12 | 0 | 25 | 2.26 | 25.00 | 27.09 |
| Mortar | 50 | 80 | 2 | 10.30 | 127.45 | 89.42 |
The ideal projectile model neglects air resistance, which is a reasonable approximation for dense, slow-moving objects over short distances. For more realistic simulations, drag force — proportional to velocity (or velocity squared) — must be included. This transforms the equations into non‑linear differential equations that require numerical integration. Our calculator provides the exact analytical solution for the ideal case, giving you a solid baseline before adding complexities. For advanced applications, consider coupling this tool with computational fluid dynamics (CFD) or using it to validate numerical solvers.
The principles of projectile motion also extend to orbital mechanics, where gravity is no longer uniform, and to relativistic regimes where speeds approach the speed of light. But for everyday terrestrial applications, the classical Newtonian framework remains exceptionally accurate and intuitive.