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Projectile Motion Calculator

Calculate range, maximum height and flight time for a projectile launched at an angle.

Formula and sources updated · Data sources · 1View algorithm, constants and default inputs (JSON)

Enter your values

Horizontal range 63.710m
Results

Horizontal range (m)

Horizontal range (m)Maximum height (m): 15.93; Horizontal range (m): 31.86 / Horizontal range (m): 0.00; Launch height (m): 0.00 / Horizontal range (m): 1.33; Launch height (m): 1.30 / Horizontal range (m): 2.65; Launch height (m): 2.54 / Horizontal range (m): 3.98; Launch height (m): 3.73 / Horizontal range (m): 5.31; Launch height (m): 4.87 / Horizontal range (m): 6.64; Launch height (m): 5.95 / Horizontal range (m): 7.96; Launch height (m): 6.97 / Horizontal range (m): 9.29; Launch height (m): 7.94 / Horizontal range (m): 10.62; Launch height (m): 8.85 / Horizontal range (m): 11.95; Launch height (m): 9.71 / Horizontal range (m): 13.27; Launch height (m): 10.51 / Horizontal range (m): 14.60; Launch height (m): 11.25 / Horizontal range (m): 15.93; Launch height (m): 11.95 / Horizontal range (m): 17.25; Launch height (m): 12.58 / Horizontal range (m): 18.58; Launch height (m): 13.16 / Horizontal range (m): 19.91; Launch height (m): 13.69 / Horizontal range (m): 21.24; Launch height (m): 14.16 / Horizontal range (m): 22.56; Launch height (m): 14.57 / Horizontal range (m): 23.89; Launch height (m): 14.93 / Horizontal range (m): 25.22; Launch height (m): 15.24 / Horizontal range (m): 26.55; Launch height (m): 15.49 / Horizontal range (m): 27.87; Launch height (m): 15.68 / Horizontal range (m): 29.20; Launch height (m): 15.82 / Horizontal range (m): 30.53; Launch height (m): 15.90 / Horizontal range (m): 31.86; Launch height (m): 15.93 / Horizontal range (m): 33.18; Launch height (m): 15.90 / Horizontal range (m): 34.51; Launch height (m): 15.82 / Horizontal range (m): 35.84; Launch height (m): 15.68 / Horizontal range (m): 37.16; Launch height (m): 15.49 / Horizontal range (m): 38.49; Launch height (m): 15.24 / Horizontal range (m): 39.82; Launch height (m): 14.93 / Horizontal range (m): 41.15; Launch height (m): 14.57 / Horizontal range (m): 42.47; Launch height (m): 14.16 / Horizontal range (m): 43.80; Launch height (m): 13.69 / Horizontal range (m): 45.13; Launch height (m): 13.16 / Horizontal range (m): 46.46; Launch height (m): 12.58 / Horizontal range (m): 47.78; Launch height (m): 11.95 / Horizontal range (m): 49.11; Launch height (m): 11.25 / Horizontal range (m): 50.44; Launch height (m): 10.51 / Horizontal range (m): 51.76; Launch height (m): 9.71 / Horizontal range (m): 53.09; Launch height (m): 8.85 / Horizontal range (m): 54.42; Launch height (m): 7.94 / Horizontal range (m): 55.75; Launch height (m): 6.97 / Horizontal range (m): 57.07; Launch height (m): 5.95 / Horizontal range (m): 58.40; Launch height (m): 4.87 / Horizontal range (m): 59.73; Launch height (m): 3.73 / Horizontal range (m): 61.06; Launch height (m): 2.54 / Horizontal range (m): 62.38; Launch height (m): 1.30 / Horizontal range (m): 63.71; Launch height (m): 0.00Launch height (m)0.05.811.717.50.031.963.7Horizontal range (m)Maximum height (m): 15.93 (31.86)Horizontal range (m): 63.71

About this calculator

A projectile's horizontal and vertical motions are independent, which is the insight that makes the maths tractable: gravity acts only vertically while horizontal velocity stays constant. Range therefore peaks at 45 degrees and falls away symmetrically either side.

Worked example

calculated with the default values below

Launch speed 25 m/s · Launch angle 45 degrees · Launch height 0 m · Gravity 9.81 m/s² Horizontal range 63.710 m · Maximum height 15.928 m · Total flight time 3.604 s · Time to the highest point 1.802 s · Impact speed 25.000 m/s · Impact angle 45.00 degrees · Optimal launch angle 45.0 degrees

Method and assumptions

Formula basis and scope

The page resolves launch speed into horizontal and vertical components and solves height + vertical velocity × time − gravity × time² / 2 = 0 for landing time.

Assumes constant gravity, a ground-level landing plane and no air resistance. Launch height is above the landing plane. Gravity is clamped to at least 0.1 m/s² by this implementation.

Data sources

Disclaimer

These calculators give general conversion results. Rounding and the precision of the value you enter both affect the output, so treat displayed decimals as indicative rather than exact.

Maintained by: CalcDelta Formula and sources updated ·

Implementation source

This record describes the implementation used by this page. It is not independent validation. External references, when available, support only the stated formula or definition; estimates and custom rules retain their own assumptions.

View algorithm, constants and default inputs (JSON)

Formula and variables

R = v cosθ × (v sinθ + √(v² sin²θ + 2gh₀)) ÷ g; t = (v sinθ + √(v² sin²θ + 2gh₀)) ÷ g

v
Launch speed
θ
Launch angle (°)
h₀
Initial height
g
Gravity
t
Flight time
R
Horizontal range

Assumes ideal projectile motion with constant gravity, no air resistance, and level landing surface except for entered launch height. At zero launch speed, an optimal-angle result is undefined.

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Frequently asked questions

Why does range peak at a 45 degree launch angle?

With no launch height, range equals v squared times sin(2θ) divided by g, which is largest when sin(2θ) is 1, at θ equals 45 degrees. The default 25 m/s launch then flies about 63.7 m. Raise the angle and vertical time grows while horizontal speed falls, so the two trade off symmetrically; only adding launch height shifts the true optimum below 45.

Next steps

After this result, you may want to…