Compute projectile motion parameters: maximum height, horizontal range, time of flight, impact velocity, and apex coordinates. Visualize the parabolic trajectory, launch point, apex, and landing point.
Projectile motion describes the curved path (trajectory) of an object launched into the air, influenced only by gravity and initial velocity (neglecting air resistance). The motion is a combination of horizontal uniform motion and vertical uniformly accelerated motion. The ballistic trajectory is a parabola, mathematically derived from Newton's laws. This calculator provides exact analytic solutions for maximum height, range, flight time, and impact conditions, essential for ballistics, sports science, and engineering design.
Equations of motion (constant gravity):
x(t) = v₀·cosθ · t | y(t) = y₀ + v₀·sinθ · t – ½·g·t²
Time to apex: tpeak = (v₀·sinθ)/g | Max height: H = y₀ + (v₀²·sin²θ)/(2g)
Range (when y=0): R = (v₀·cosθ/g)·(v₀·sinθ + √((v₀·sinθ)² + 2g·y₀))
Our algorithm solves projectile motion analytically. First, velocity components are computed: vₓ = v₀·cosθ, vᵧ = v₀·sinθ. The time to maximum height occurs when vertical velocity becomes zero: tup = vᵧ/g. Maximum height = y₀ + vᵧ·tup – ½·g·tup². Flight time T is found by solving the quadratic: y₀ + vᵧ·T – ½·g·T² = 0, taking the positive root. Horizontal range = vₓ·T. Impact velocity magnitude = √(vₓ² + (vᵧ – g·T)²) and impact angle = arctan(|vᵧ – g·T| / vₓ). The apex coordinates are (vₓ·tup, y₀ + vᵧ·tup – ½·g·tup²). All calculations respect real-number validity; if initial velocity is zero or negative, a warning is shown.
| Scenario | v₀ (m/s) | θ (°) | y₀ (m) | Range (m) | Max Height (m) | Flight Time (s) |
|---|---|---|---|---|---|---|
| Flat ground, 45° | 50 | 45 | 0 | 254.84 | 63.71 | 7.21 |
| High launch, 30° | 100 | 30 | 20 | 916.50 | 147.40 | 10.58 |
| Moon gravity (g=1.62) | 40 | 30 | 0 | 855.33 | 123.46 | 24.69 |
| Vertical shot (θ=90°) | 30 | 90 | 0 | 0 | 45.87 | 6.12 |
A soccer player kicks a ball from 25 meters away, with goal height 2.44 m. Initial height is 0.2 m. Using this calculator, the player tests different angles: 30°, 40°, 50° at v₀=25 m/s. The calculator shows that 40° yields a peak height of 13.1 m and range 65.4 m – clearing the wall easily. The interactive graph visualizes the ball clearing the defensive wall and dipping into the goal. This real‑world application demonstrates how projectile analysis informs athletic strategy and game design.
While our model assumes ideal conditions (no drag, flat Earth, uniform gravity), real ballistics involve air resistance (quadratic drag), Coriolis effects for long ranges, and variable gravity. However, the ideal projectile equations serve as foundational tools for understanding trajectories. For small projectiles at moderate speeds (e.g., tennis balls, soccer kicks), the error remains under 5-10%. Military ballistics add complex drag models, but our calculator gives a first‑order approximation indispensable for students and rapid prototyping.
Quantitative note on air resistance: For a typical spherical projectile (diameter 0.1 m, mass 1 kg) launched at 50 m/s, air resistance can reduce the range by approximately 15–20% compared to the ideal parabolic flight. The effect becomes more pronounced at higher velocities and for lighter objects. Use our ideal model for educational and initial design purposes; for precision ballistics (e.g., long‑range shooting), specialized drag models are required.