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Projectile Motion Calculator

Find the range and height of a projectile.

Result

How it is calculated

The launch speed v (m/s) and angle θ (degrees above horizontal) give three results with g = 9.81 m/s²: range R = v² × sin(2θ) / g, maximum height H = v² × sin²(θ) / (2g) and flight time T = 2 × v × sin(θ) / g. The model assumes launch and landing at the same height, no air resistance and constant gravity. Results are in metres and seconds, rounded to two decimals. The angle must be between 0 and 90 degrees and the speed 0 or more; anything outside that returns no result. At 90 degrees the object goes straight up and the range is 0.

Worked example

Launch at v = 20 m/s and θ = 30 degrees. Then v² = 400. Range: 400 × sin 60° / 9.81 = 400 × 0.8660 / 9.81 = 35.31 m. Height: 400 × sin²30° / 19.62 = 400 × 0.25 / 19.62 = 5.10 m. Time: 2 × 20 × sin 30° / 9.81 = 20 / 9.81 = 2.04 s. The object peaks at 5.10 m and lands 35.31 m away after 2.04 seconds.

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Common mistakes

  • Entering the angle in radians. The field is degrees, so 45 degrees is 45, not 0.785.
  • Entering the angle measured from the vertical. θ is measured from the ground, so a steeper angle gives more height but, above 45 degrees, less range.
  • Expecting a longer range from 60 degrees than from 45. Complementary angles such as 30 and 60 give the same range (35.31 m at 20 m/s) while 45 gives the maximum, 40.77 m.
  • Using the results for a launch from a cliff or a raised platform. The formulas assume the landing is at launch height.
  • Entering speed in km/h. Convert to m/s first by dividing by 3.6.

About this calculator

This calculator solves classic projectile motion, splitting an object's launch velocity into horizontal and vertical components to find range (R = v²sin(2θ)/g), maximum height (H = v²sin²θ/2g), and flight time (T = 2v·sinθ/g) — assuming ideal conditions with no air resistance.

It's a standard physics-class problem for finding where a launched ball, cannonball, or thrown object lands, and the same underlying math is used informally in sports like golf and basketball to understand how launch angle trades off against distance and hang time.

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Comments (1)

  • Editorial team

    Notice how fast the numbers grow with speed. Doubling the launch speed from 20 to 40 m/s at 30 degrees quadruples both range and height, from 35.31 m to 141.25 m for range, while flight time only doubles, from 2.04 s to 4.08 s. Real throws fall short of these figures because of air drag, and the lighter or faster the object, the bigger the gap. What are you launching, and does the result match what you measure?

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