Projectile Motion Calculator — Range, Height and Flight Time
A projectile moves under gravity alone: its horizontal velocity never changes, while its vertical velocity is accelerated downward at g. This calculator splits the launch velocity into horizontal and vertical components and works out the flight time, the range and the highest point of the trajectory.
Projectile problems appear in mechanics courses everywhere because they combine two independent one-dimensional motions into one curved path. Air resistance is ignored, which is a good model for dense objects at modest speeds.
Use the calculator
Enter launch speed and angle. Launch height and g are optional.
The formula
Key relations: vx = v₀ cos θ (constant), vy = v₀ sin θ − gt. Range R = vx × T, maximum height H = y₀ + (v₀ sin θ)² / 2g, and on flat ground R = v₀² sin 2θ / g — which is largest at θ = 45°.
Worked example
Worked example: A ball is launched at 20 m/s at 45° from ground level. Components: vx = vy = 20 × cos 45° ≈ 14.14 m/s. Flight time T = 2 × 14.14 / 9.81 ≈ 2.88 s. Range R = 14.14 × 2.88 ≈ 40.8 m. Maximum height H = 14.14² / (2 × 9.81) ≈ 10.2 m.
Frequently asked questions
What launch angle gives the maximum range?
On flat ground with no air resistance, 45 degrees. The range formula R = v₀² sin 2θ / g is maximised when sin 2θ = 1, which happens at θ = 45°.
Why is horizontal velocity constant in projectile motion?
Because gravity acts only vertically. With no horizontal force there is no horizontal acceleration, so the horizontal component of velocity stays at its launch value for the whole flight.
Does the mass of the projectile affect the range?
No. In the no-air-resistance model, mass cancels out: flight time, range and height depend only on launch speed, angle, height and g.