Science calculator

Projectile motion.

Enter a launch speed and angle to find the horizontal range, maximum height and time of flight. This level-ground model neglects air resistance; its assumptions are stated below.

Launch speed u

Launch angle θ

Gravitational field strength g

Horizontal range R level ground

—

horizontal distance on level ground

Maximum height H: —
Time of flight T: —

Distance unit

Display precision

R = \frac{u^{2} \sin (2 θ)}{g}

H = \frac{u^{2} {\sin (θ)}^{2}}{2 g}

T = \frac{2 u \sin (θ)}{g}

u launch speed · θ angle above the horizontal · g gravitational field strength
R horizontal range · H maximum height above launch level · T time of flight

Assumptions

The projectile lands at the same vertical level from which it was launched.
Air resistance, lift, wind and spin are neglected.
The gravitational field is uniform and acts vertically downwards.
The projectile is treated as a point particle; the launch speed and angle are its values immediately after release.
The launch angle is measured above the horizontal and must be between 0° and 90°.

How it works

Projectile motion separates into independent horizontal and vertical components. The horizontal velocity u cos(θ) stays constant, while the vertical velocity u sin(θ) changes under the acceleration −g.

On level ground, the upward and downward parts of the flight take equal times. Substituting that flight time into horizontal distance gives the range formula. The greatest height occurs when vertical velocity reaches zero. These shortcuts do not apply unchanged when launch and landing heights differ.

Results use up to three significant figures by default (adjustable above), without padding with trailing zeroes. Calculation uses full browser number precision before the display is rounded.

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