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Projectile Motion Calculator

Find a projectile's max height and range.

How it is calculated

Maximum height is H = v²·sin²θ / (2g) and horizontal range is R = v²·sin(2θ) / g. Here v is the launch speed in m/s, θ the launch angle in degrees above the horizontal, and g = 9.81 m/s². Both results are in metres, rounded to two decimals.

The speed must be above zero and the angle must be between 0 and 90 degrees. The formulas assume the object starts and lands at the same height on level ground, with no air resistance and no wind, so the range is not valid for a launch from a cliff or a platform. At 90° the height is at its largest and the range is 0.00 m, because the object goes straight up and comes back down. For a given speed the range is greatest at 45°.

Worked example

A ball is thrown at 20 m/s at 30°. sin 30° = 0.5, so sin² = 0.25. H = 20² × 0.25 / (2 × 9.81) = 100 / 19.62 = 5.10 m. The range uses sin 60° = 0.866: R = 400 × 0.866 / 9.81 = 35.31 m. At 60° instead, the range is the same 35.31 m, but the height rises to 15.29 m, because 30° and 60° are complementary angles.

Try this example →

Common mistakes

  • Entering the speed in km/h: the formulas use m/s, so 72 km/h must be entered as 20.
  • Measuring the angle from the vertical instead of the horizontal: 30° from the vertical is 60° here, which gives a very different height and range.
  • Using the result for a launch from a height, such as a ball thrown off a building: the range here assumes it lands at the launch level.
  • Expecting real-world distances: with drag the true height and range are smaller, especially for light or fast objects.

About this calculator

This calculator applies the kinematic equations of projectile motion under constant gravity to find the maximum height, h = (v²·sin²θ)/(2g), and the horizontal range on level ground, R = (v²·sin2θ)/g, of an object launched at speed v and angle θ, ignoring air resistance. These equations come from decomposing the launch velocity into horizontal and vertical components and applying standard equations of motion to each independently.

Physics students use it to solve classic mechanics problems, ballistics analysts and military applications use the same equations for artillery trajectory estimates, and sports scientists analyze it for basketball shots, golf drives, and javelin throws where launch angle directly affects distance. Game developers modeling realistic projectile physics and hobbyists working with model rockets or trebuchets also reach for this calculator to predict how far and how high something will fly.

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