Projectile Range calculator

Enter launch speed and angle. The calculator computes range Z = v₀² sin(2α)/g on flat ground. 20 m/s at 45° is about 40.8 m, the farthest flight at that v₀ when launch and landing sit on the same level.

Result

Initial speed v₀, Angle α (degrees) and g (default 9.81). The result shows up here.

How it works

Range on flat ground - maximum at α = 45° (no air resistance).

Range on level ground is Z = v₀² sin(2α) / g. At 20 m/s and 45°, sin(2α) = 1, so Z = 400 / 9.81 ≈ 40.8 m. At 30° and 60° sin(2α) is the same, so Z is the same, about 35.3 m, even though the arcs look different: 30° stays flat, 60° climbs higher. At 28 m/s and 40° you get about 78.7 m.

Fields: v₀ in m/s or the labeled unit, α in degrees, empty g = 9.81 m/s². Z is in meters. The formula assumes the same launch and landing height and no air drag. 45 in the angle field is 45°, not π/4 typed as 0.785. A comma and a period are the same v₀: 20,5 and 20.5.

Zero degrees gives Z = 0: there is no upward component and you do not leave the ground. α = 90° also gives Z = 0, because that is a vertical throw and the flight returns at your feet. Zero m/s gives Z = 0. v₀ must be a number before 40.8 m can appear.

Compute H_max next door: at 20 m/s and 45° the peak is about 10.2 m, not 40.8 m. A launch from height h₀ is on the full-projectile page. A horizontal throw from an edge is α = 0° and a nonzero h₀, a different card.

The largest Z on the flat, with no drag, is at 45°. Two angles that add to 90° give the same range. v₀ is squared: twice the speed is four times Z, not two.

Type 20 and 45, leave g empty, click Calculate, and match about 40.8 m. Then 20 and 30, then 20 and 60: the same Z, a different shape. The header symbol does not kick the ball.

Peak of the arc: H_max. Launch from a tower: full projectile.

How to use

In the first field enter v₀ in m/s, for example 20. That is speed at launch, not a mean over the whole flight.
In the second field enter angle α in degrees, for example 45. The field wants degrees, not radians. 45 is 45°, not 0.785.
Leave g blank for Earth, or type 9.81. Click Calculate. 20 m/s and 45° give about 40.8 m. 20 m/s and 30° give about 35.3 m.
Launch and landing are on the same level. Zero degrees or 90° give Z = 0. Zero v₀ does too. A tower needs the full-projectile page.
The peak of the arc lives on the H_max page. At 20 m/s and 45° that is about 10.2 m. Here it stays range Z on flat ground.

Formula

Z = v₀² sin(2α) / g

H_max = v₀² sin²(α) / (2g)

Same-level launch and landing, no air resistance.

Z, v₀, and α on level ground

Range in this calculator assumes the same launch and landing height. 20 m/s at 45° gives Z ≈ 40.8 m, farthest at that v₀ with no drag.

Z: Range in meters: Z = v₀² sin(2α)/g. At 30° and 60° sin(2α) matches, so Z ≈ 35.3 m at 20 m/s.
v₀: Launch speed, usually m/s. It is squared: twice v₀ is four times Z. Zero m/s gives Z = 0.
α: Angle in degrees, not radians. 45 in the field is 45°. 0° and 90° give Z = 0: no upward part, or no horizontal part.
g: Gravitational acceleration. Blank field = 9.81 m/s². It sits in the denominator of Z.
H_max: Peak shown as a helper. At 20 m/s and 45° that is about 10.2 m, not 40.8 m. The main result here is Z.

Real-life examples

Example 1

v₀ = 20 m/s, α = 45°.

Example 2

v₀ = 20 m/s, α = 30°.

Example 3

v₀ = 20 m/s, α = 60°: same Z as 30°.

Example 4

v₀ = 28 m/s, α = 40°.

Example 5

v₀ = 25 m/s, α = 35°.

Example 6

v₀ = 18 m/s, α = 40°.

Example 7

v₀ = 20 m/s, α = 45°, g = 1.62 m/s².

Example 8

v₀ = 12 m/s, α = 15°.

Ways to use this calculator

You check how far a ball goes at 20 m/s and 45°.
You compare 30° and 60° at the same v₀.

Frequently asked questions

At what angle is range largest?

At 45°, when launch and landing are on the same level and there is no air drag. At 20 m/s that is about 40.8 m.

Why do 30° and 60° give the same Z?

Because sin(2·30°) = sin(60°) = sin(2·60°) = sin(120°). Same sin(2α), same range, about 35.3 m at 20 m/s, different arc shape.

How many meters at 20 m/s and 45°?

Z = 400 / 9.81 ≈ 40.8 m. H_max at 45° is about 10.2 m, computed next door. At 28 m/s and 40°, Z ≈ 78.7 m.

Can I launch from a tower?

This formula assumes Δh = 0. From a height, open full projectile or horizontal throw. Those pages take h₀.

Which units do I type?

v₀ in m/s on the label, α in degrees, g in m/s². Z in meters. Empty g = 9.81.

What if v₀ = 0 or α = 0°?

Z = 0. Zero speed never leaves. Zero degrees has no upward component. 90° is a vertical throw, Z is 0 again.

Is 45 in the angle field radians?

No. The field wants degrees. 45 is 45°, not π/4 typed as 0.785. There are no radians here.

Where is H_max?

On the projectile-height page. Here the main result is Z. At 20 m/s and 45° the peak is about 10.2 m, not 40.8 m.

Is there air drag?

No. Classroom vacuum model. In air, 45° is usually no longer the best angle.

Does a comma in 20.5 work?

Yes. 20,5 and 20.5 are the same v₀. The calculator does not require a period.

Knowledge sources

The formula is the school one. Units follow SI; NIST SP 330 and BIPM define the measures, not your result.

Page updated in 2026.