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R = v cos θ × t

Projectile motion calculator

Range, apex height, flight time and impact speed from a launch speed and angle, with an optional release height above the ground.

Short answer: 20 m/s at 45° from level ground travels 40.79 m, peaks at 10.20 m after 1.44 s, and lands 2.88 s after launch. Both distance and height are the same because sin 45° and cos 45° are equal.

The two components

Splitting the launch into v_x = v cos θ and v_y = v sin θ is the whole method: horizontal motion is constant speed, vertical motion is constant acceleration, and the two are joined only by the clock. The page prints both components so the split is visible rather than assumed.

Worked example

A football kicked at 25 m/s and 35°: v_y = 14.34 m/s, flight 2 × 14.34 ÷ 9.80665 = 2.925 s, range 20.48 × 2.925 = 59.9 m and apex 10.49 m. Air resistance would cut that by roughly a fifth in a real match, which is why the assumption is printed under the result.

Projectile questions

What does the calculator assume?

No air resistance and a flat ground at the release height unless you enter one. Those are the two assumptions the school formulas carry, and stating them matters because a real ball falls short of the ideal range — sometimes a third short.

Which angle gives the greatest range?

45° from level ground. 30° and 60° give the same range with different apex heights, which is the quickest way to check the arithmetic: the page reports both, and they match to the last digit.

How does a release from a height change things?

It lengthens the flight, so the range grows and the impact speed is higher than the launch speed. The time is solved as (v_y + √(v_y² + 2gh)) ÷ g rather than the level-ground 2v_y ÷ g, which would understate a throw off a wall.

Why is the impact speed not the same as the launch speed?

It is the same only for a level launch, where the vertical component returns to its starting magnitude. Released from a height, the projectile has gained speed on the way down — that is the same energy relationship the energy calculator uses.

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Speed, distance and time calculatorConstant-speed motion.Energy calculatorKinetic and potential energy.Force calculatorF = ma and weight.