What is Projectile Range?

Projectile range is the horizontal distance a projectile travels before hitting the landing level. It depends on launch speed, launch angle, initial height, and gravity. At equal launch and landing heights, range is maximized at a 45° launch angle, but with non-zero launch height the optimal angle shifts lower.

Use this calculator to determine range, time of flight, and peak height for any launch speed, angle, and initial height. It is ideal for physics problems, ballistics estimates, sports trajectory analysis, and engineering tests where accurate flight distances are required.

Projectile Motion Formulas

t_{flight} = \frac{v_0 \sin\theta + \sqrt{v_0^2 \sin^2\theta + 2 g h_0}}{g}

Time of flight with launch height h₀, launch speed v₀, angle θ, and gravity g.

R = v_0 \cos\theta \times t_{flight}

Horizontal range equals horizontal velocity times time of flight.

h_{max} = h_0 + \frac{v_0^2 \sin^2\theta}{2g}

Maximum height measured from the landing reference level.

R_{level} = \frac{v_0^2 \sin(2\theta)}{g} \quad (h_0 = 0)

Simplified range when launch and landing heights are the same.

How to Use the Projectile Range Calculator

Enter launch speed: Choose units m/s, km/h, mph, or ft/s.
Set launch angle: Angle above horizontal in degrees.
Add launch height: Height above landing level (meters).
Adjust gravity if needed: Default 9.81 m/s², change for other planets.
Click Calculate: Get range, time of flight, and maximum height with instant results.

Optimizing Range

For launches where start and landing heights are equal, maximum range occurs at 45°. If the launch point is higher, optimal angles are typically below 45° because gravity acts longer on higher arcs. Increasing launch speed, reducing gravity, or lowering drag (not modeled here) increases range.

Practical Example

Scenario: v₀ = 30 m/s, θ = 40°, h₀ = 2 m, g = 9.81 m/s².

t = (30·sin40° + √((30·sin40°)² + 2·9.81·2)) / 9.81 ≈ 2.07 s
Range R = 30·cos40° · 2.07 ≈ 47.6 m
Max height hmax = 2 + (30²·sin²40°)/(2·9.81) ≈ 10.4 m

Result: About 47.6 meters range, 2.07 seconds flight time, and 10.4 meters peak height.

Does this calculator include air resistance?

No. It assumes ideal projectile motion without drag. Use it for vacuum or low-drag approximations.

Which angle gives maximum range?

If launch and landing heights are equal, 45° maximizes range. With launch height above target, optimal angles are typically below 45°.

Can I change gravity?

Yes. Enter any gravity value to model trajectories on Earth, Moon, Mars, or custom environments.

What units should I use?

Use consistent speed units; the calculator converts internally. Output distances are shown in meters and times in seconds.

Is launch height required?

No. Leave height at 0 for ground-level launches. Enter a positive height for elevated launches to extend range.

Projectile Range Calculator – Projectile Motion

Projectile Range Calculator

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