Projectile Range Calculator

Projectile motion is one of the fundamental concepts in classical physics. A projectile is any object launched into the air that moves only under the influence of gravity (ignoring air resistance). Understanding its trajectory lets you calculate how far it travels, how high it goes, and how long it stays airborne.

Key Assumptions

This calculator uses ideal projectile motion, which assumes:

• No air resistance
• Flat ground (launch and landing at the same height)
• Constant gravitational acceleration (g = 9.81 m/s² or 32.2 ft/s²)

The Formulas

Given an initial speed v₀ and launch angle θ:

Horizontal Range: R = (v₀² × sin(2θ)) / g

Maximum Height: H = (v₀² × sin²(θ)) / (2g)

Time of Flight: T = (2 × v₀ × sin(θ)) / g

Horizontal Velocity (constant throughout): vₓ = v₀ × cos(θ)

Vertical Velocity (at launch): vᵧ = v₀ × sin(θ)

Optimal Launch Angle

The range is maximised at exactly 45° when the launch and landing points are at the same height. At this angle, sin(2θ) = sin(90°) = 1, which gives the maximum possible value. Angles the same distance either side of 45° produce equal ranges, so 30° and 60° cover identical ground.

That symmetry breaks the moment the ground does. Throwing downhill, or off a cliff, or from shoulder height onto a flat field, all shift the best angle below 45°, and the bigger the drop the flatter the optimum gets. A shot putter releases the ball about 2.1 m up at around 13.5 m/s, which drops the ideal angle to roughly 42°.

Watch a real competition, though, and the good ones throw nearer 34°. The extra flattening has nothing to do with the maths above: a human simply cannot push as hard upward as forward, so the speed you lose climbing past the low thirties costs more range than the angle gains. It is a useful reminder that the optimum in a physics model is the optimum for that model alone.

Practical Examples

A ball thrown at 15 m/s (54 km/h) at 45°:

• Range: (15² × 1) / 9.81 ≈ 22.9 metres
• Max height: (15² × 0.5) / (2 × 9.81) ≈ 5.7 metres
• Time of flight: (2 × 15 × 0.707) / 9.81 ≈ 2.16 seconds

In imperial units, the same ball at 49.2 ft/s at 45°:

• Range ≈ 75.2 feet
• Max height ≈ 18.8 feet

Real-World Applications

• Sports: optimising the angle for throwing a javelin, shot put, long jump
• Engineering: ballistics, irrigation sprinklers, water jets
• Gaming and simulation: calculating realistic trajectories
• Physics education: understanding vectors and gravity

Note on Air Resistance

Real-world projectiles are slowed by air drag, especially at high speeds. A baseball at 40 m/s will travel noticeably shorter than this calculator predicts. For precision applications (artillery, aerospace), full ballistic models including drag coefficients are required.