Projectile Motion Calculator — Range, Height & Time of Flight

The Equations of Projectile Motion

Projectile motion separates cleanly into two independent components:

• Horizontal: x = v₀cos(θ) × t — constant velocity, no acceleration
• Vertical: y = h₀ + v₀sin(θ) × t − ½g × t² — uniformly accelerated by gravity

The key insight is that horizontal and vertical motions are independent. Horizontal velocity never changes (ignoring air resistance). Vertical velocity changes at rate g = 9.81 m/s² downward throughout the flight.

Worked Examples

How to Use the Calculator

1. Open the projectile motion calculator.
2. Enter initial velocity in m/s. This is the total launch speed.
3. Drag the angle slider to set the launch angle from 0° to 90°. The trajectory diagram updates in real time.
4. Optionally enter an initial height to simulate launches from an elevated platform or cliff.
5. Adjust gravity to simulate projectile motion on other planets (Moon = 1.62 m/s², Mars = 3.72 m/s²).
6. Read off range, maximum height, time of flight, time to apex, and velocity components.

The 45° Optimal Angle

For flat ground with no air resistance, a 45° launch angle gives the maximum range. Here is why: the range formula is R = v₀² × sin(2θ) / g. The function sin(2θ) is maximized when 2θ = 90°, i.e. θ = 45°.

Complementary angles (e.g. 30° and 60°) give identical range because sin(2×30°) = sin(60°) = sin(120°) = sin(2×60°). Verify this in the calculator: enter the same speed at 30° and 60° — the ranges match.

Advanced Workflows

Launching from elevation

When h₀ > 0, the optimal angle for maximum range shifts below 45°. A cliff launch at h₀ = 10 m and v₀ = 20 m/s achieves maximum range at about 37°, not 45°. Experiment in the calculator by varying the angle while watching the range value.

Projectile motion on other planets

Set g = 1.62 m/s² (Moon) and compare the range with Earth (9.81 m/s²). The same throw at 45° goes 6.05 times farther on the Moon. Astronauts on the Apollo missions could throw objects enormous distances precisely because of this reduced gravity.

Common Questions

Why does the range not depend on the mass of the projectile?

In the absence of air resistance, all objects fall at the same rate regardless of mass (Galileo's equivalence principle). Mass cancels out of the equations entirely, so a golf ball and a cannonball launched at the same speed and angle travel the same range.

How does air resistance change things?

Air resistance significantly reduces range and shifts the optimal angle below 45°. For a real golf ball, the optimal angle is about 10–12° because dimples create lift but drag reduces the benefit of high-angle shots. This calculator ignores drag for simplicity.