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Projectile Motion Calculator

Find projectile range, max height, and flight time from launch velocity and angle. All three outputs shown with working. Assumes ground-level launch, no drag.

Result

40.77 m

A projectile launched at 20 m/s at 45° travels 40.77 m, reaching 10.19 m high

Quick Answer

A projectile motion calculator uses launch velocity (v) and launch angle (θ) to compute three outputs: range = v²sin(2θ)/g, maximum height = v²sin²θ/(2g), and time of flight = 2v sinθ/g. All formulas use g = 9.81 m/s². For v = 20 m/s at 45°, range = 400 × sin(90°)/9.81 = 40.77 m, max height = 400 × sin²(45°)/19.62 = 10.19 m, time = 2 × 20 × sin(45°)/9.81 = 2.89 s.

Projectile Motion: Three Outputs From Two Inputs Explained

A projectile motion calculator takes two inputs, launch velocity and launch angle, and returns the three key kinematic results for a projectile launched from ground level with no air resistance. Range is the horizontal distance at landing. Maximum height is the peak altitude above the launch point. Time of flight is the total duration from launch to landing. These three quantities are fully determined by the two inputs under the standard assumption set: flat ground, no air drag, constant gravitational acceleration of 9.81 m/s².

The Projectile Motion Formulas and Their Variables

Range: R = v² × sin(2θ) ÷ g Max height: H = v² × sin²(θ) ÷ (2g) Time of flight: T = 2v × sin(θ) ÷ g

  • Range: R = v² × sin(2θ) ÷ g
  • Maximum height: H = v² × sin²(θ) ÷ (2g)
  • Time of flight: T = 2v × sin(θ) ÷ g
  • Gravitational acceleration: g = 9.81 m/s²

How To Use the Projectile Motion Calculator Step By Step

Inputs

  • Launch velocity (m/s): the speed of the projectile at the instant of launch
  • Launch angle (degrees): the angle of the velocity vector above the horizontal at launch; must be between 0° and 90°

Steps

  1. Enter launch velocity in m/s.
  2. Enter launch angle in degrees.
  3. Read range, maximum height, and time of flight from the output.
  4. Note that all three results are valid only for the no-drag, flat-ground, ground-level-launch assumption set.

Projectile Motion Worked Example: 30 m/s at 60 Degrees

A projectile launched at 30 m/s at an angle of 60° above the horizontal.

  1. Calculate Range: R = 30² × sin(120°) ÷ 9.81 = 900 × 0.8660 ÷ 9.81 = 79.45 m.
  2. Calculate Max Height: H = 30² × sin²(60°) ÷ 19.62 = 900 × 0.7500 ÷ 19.62 = 34.40 m.
  3. Calculate Time of Flight: T = 2 × 30 × sin(60°) ÷ 9.81 = 60 × 0.8660 ÷ 9.81 = 5.30 s.

Range = 79.45 m, max height = 34.40 m, time of flight = 5.30 s.

When Projectile Motion Formulas Apply and When They Do Not

Use this calculator for physics problems, classroom exercises, or initial engineering estimates where idealised projectile behaviour is acceptable. The results are accurate when air resistance is negligible (low-speed, dense, compact objects over short ranges) and when the launch and landing surfaces are at the same elevation. Do not use these results for high-speed ballistics or launches from an elevated position.

Assumptions

  • Launch and landing at the same ground level.
  • No air resistance (drag coefficient = 0).
  • Constant g = 9.81 m/s² throughout the trajectory.
  • Launch angle between 0° and 90°.

Limitations

  • Does not account for air drag, spin, wind, or Coriolis effects.
  • Does not calculate landing coordinates or trajectory coordinates at intermediate times.
  • Does not handle launches from an elevated launch point (initial height > 0).

In Practice

The most common mistake when using projectile motion formulas is applying them to situations where air resistance is significant. A golf ball, for example, travels about 10–15% shorter than the no-drag formula predicts because of aerodynamic drag. A dense steel ball bearing over a short distance matches the formula closely. When in doubt, compare the drag force (½ρCdAv²) to the gravitational force (mg) at the launch velocity; if the ratio is above 5–10%, the no-drag result is a rough estimate only.

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Frequently Asked Questions: Projectile Motion Calculator

What angle gives the maximum range?

A launch angle of 45° maximises range for a ground-level launch with no air resistance. At 45°, sin(2θ) = sin(90°) = 1, which is its maximum value, so range = v²/g. For any other angle, range is less than v²/g.

Why do 30° and 60° launches give the same range?

Both angles produce the same sin(2θ) value: sin(60°) = sin(120°) = 0.866. This is because sin(2θ) = sin(180° − 2θ), which holds whenever two angles sum to 90°. The 60° launch reaches a higher peak and takes longer in the air, but covers the same horizontal distance as the 30° launch.

What is g and why is it 9.81?

g is the standard gravitational acceleration at Earth's surface, defined as 9.80665 m/s² by international standard. The value 9.81 m/s² is the two-decimal approximation used in most physics problems. It varies slightly with latitude and altitude, but 9.81 is accurate to within 0.5% for most surface-level calculations.

Can I use this calculator in feet and miles per hour?

No. The formulas use SI units: velocity in m/s and distance in metres. To convert: 1 mph = 0.4470 m/s; 1 foot = 0.3048 m. Convert inputs to m/s, run the calculation, then convert outputs to feet if needed.

What happens if I enter 90° as the launch angle?

At 90° (vertical launch), range = v²sin(180°)/g = 0 (the projectile goes straight up and lands on the launch point), max height = v²/2g, and time of flight = 2v/g. These are the correct results for a vertical launch under ideal conditions.

Sources

Last updated: . Reviewed for accuracy against the formula shown above.