Projectile Motion Calculator: Range, Height, and Trajectory Plot

Solve 2D projectile motion for range, maximum height, time of flight, and impact speed from launch speed, angle, and initial height. The trajectory is plotted and the value of g is named.

Unit system

Metric (m, m/s) US customary (ft, ft/s)

Launch speed v0

Initial speed. Required.

Launch angle

Measured from the horizontal. Required.

Launch height h

Height above the landing point. Blank counts as 0.

Gravity g

Always in m/s², independent of the length unit.

Gravity preset

Writes a value into the g field above. A larger g gives a shorter, lower flight. This is not a single 1/g rule: only the time to apex scales as 1/g, and the other outputs also depend on the launch height.

Angle slider 45 °

Drag to set the launch angle. After the first run, the arc replots as you drag.

Angle is measured from the horizontal. Launch height is the height above where the projectile lands. The model is drag-free: constant gravity and a vacuum parabola.

Range

Enter launch speed and angle

Run the tool to read the outputs and draw the arc.

The trajectory arc draws here after you run the tool.

Results
Quantity	Value	Unit

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The value of g is stated, and it shapes every result

The value of g drives the result, but not by a single 1/g rule. From level ground (launch height 0), the time of flight, range, and maximum height each scale as 1/g, while the impact speed equals the launch speed and the impact angle equals the launch angle, so both are independent of g. Once the launch height is above 0, only the time to apex, v0y/g, keeps the clean 1/g form; the time of flight, range, and maximum height bend away from it because g also enters through the fall from that height, and the impact speed sqrt(v0² + 2 g h) rises with g. This calculator uses g = 9.81 m/s² by default; standard gravity (9.80665) and the common textbook value (9.8) are one click away in the preset, and g stays editable. Launch height is measured above the landing point, and the model is drag-free, with no air resistance.

T = (v0y + sqrt(v0y² + 2 g h)) / g; R = v0x T; H = h + v0y² / (2 g) How?

How this is calculated

Projectile motion is two independent one-dimensional motions: horizontal motion at constant velocity and vertical motion at constant acceleration g downward. Resolve the launch speed into components v0x = v0 cos(angle) and v0y = v0 sin(angle).

Because the launch height h can be non-zero, the time of flight is the positive root of the quadratic y(t) = h + v0y t - 0.5 g t² = 0, giving T = (v0y + sqrt(v0y² + 2 g h)) / g. The range is R = v0x T. The maximum height above the landing datum is H = h + v0y² / (2 g), reached at time v0y / g. The impact speed follows from energy, v = sqrt(v0² + 2 g h), and the impact angle below the horizontal is atan2(sqrt(v0y² + 2 g h), v0x).

At h = 0 these reduce to the familiar symmetric forms T = 2 v0 sin(angle) / g, R = v0² sin(2 angle) / g, and H = v0² sin²(angle) / (2 g).

Formula: T = (v0y + sqrt(v0y² + 2 g h)) / g; R = v0x T; H = h + v0y² / (2 g)

See Sources below

How to calculate projectile motion (worked example)

Launch a ball at v0 = 20 m/s and 45 degrees from level ground (h = 0), with g = 9.81 m/s².

Components: v0x = 20 cos(45°) = 14.14 m/s, v0y = 20 sin(45°) = 14.14 m/s.
Time of flight: T = 2 (14.14) / 9.81 = 2.883 s.
Range: R = 14.14 (2.883) = 40.77 m.
Maximum height: H = 14.14² / (2 (9.81)) = 10.19 m, reached at 1.442 s.

The projectile lands at the same speed it left, 20 m/s, at 45 degrees below the horizontal, because the flight is symmetric when the launch and landing heights match.

Assumptions and model

Point mass, so shape and spin do not matter.
Drag-free flight: constant gravity and a vacuum parabola.
Constant g over the whole flight.
Flat ground at the landing datum, y = 0.
Angle measured from the horizontal, with up positive and gravity acting downward.

Sources

OpenStax University Physics Volume 1, §4.3 Projectile Motion (Example 4.7). OpenStax. Retrieved Jul 10, 2026.

How we choose and check our formulas