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Projectile Motion Calculator

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Calculate range, peak height or flight time for ideal projectile motion — enter launch speed, angle and gravity, and choose which one to solve for.

Launch conditions

Solve for
m/s
The initial speed of the projectile at launch.
°
Angle above the horizontal.
m
Height above the ground the projectile is launched from. Leave at 0 for ground level.
m
Height of the landing point, measured on the same scale as launch height. Leave at 0 for level ground.
m/s²
Standard Earth gravity is 9.81 m/s². Try 1.62 for the Moon or 3.71 for Mars.
Solving for
Range
Gravity used
9.81 m/s² (Earth standard)
Launch height
0 m
Landing height
0 m
Angle unit
Degrees
Model
No air resistance, constant gravity
Range
40.7747m
Horizontal velocity (m/s)
14.1421
Vertical velocity (m/s)
14.1421
Peak height (m)
10.1937
Flight time (s)
2.8832
Convert the angle
θ = 45° × (π ÷ 180) = 0.7854 rad
Horizontal velocity
vₓ = v₀ × cos(θ) = 20 × cos(45°) = 14.1421 m/s
Vertical velocity
v_y = v₀ × sin(θ) = 20 × sin(45°) = 14.1421 m/s
Time to peak height
t_peak = v_y ÷ g = 14.1421 ÷ 9.81 = 1.4416 s
Flight time
T = (v_y + √(v_y² + 2g(h₀ − h_land))) ÷ g = 2.8832 s
Range
R = vₓ × T = 14.1421 × 2.8832 = 40.7747 m
Result
Range = 40.7747 m

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How to use this calculator

Choose what to solve for, then enter the launch conditions — the result updates automatically as you type, with no separate calculate step.

  1. Choose "Range", "Peak height" or "Flight time" as what to solve for. All three are calculated every time; this only changes which one is the headline figure.
  2. Enter the launch speed and launch angle (measured above the horizontal).
  3. Optionally set a launch height above the ground, and a landing height for a target above or below the launch point — leave both at 0 for a level-ground launch.
  4. Optionally change gravity from Earth standard to model the Moon, Mars or another value.
  5. Review the horizontal and vertical velocity components, the other two calculated quantities, and the full step-by-step working in the summary panel below the result.

What do I need to enter?

  • Solve for — Range, Peak height or Flight time. It only picks the headline figure; the other two still appear below it.
  • Launch speed — the initial speed in m/s, from 0 to 2,000.
  • Launch angle — degrees above the horizontal, from 0 to 90.
  • Launch height — metres above the ground the projectile starts from, 0 to 10,000. Leave at 0 for ground level.
  • Landing height — metres, 0 to 10,000, on the same scale as launch height. Leave at 0 for level ground.
  • Gravity — m/s² from 0.1 to 30. The default is Earth's 9.81.

What do the results show?

  • Range, peak height or flight time — the main figure, whichever you chose to solve for, in metres or seconds.
  • Horizontal velocity and vertical velocity — the launch speed split into its two components, in m/s.
  • The other two quantities — range, peak height and flight time are all shown. Range and flight time show a dash when the landing height is above the peak.
  • Summary tab — the step-by-step working with your numbers substituted in.

The formula

Launch speed splits into two independent components — horizontal, which stays constant through the flight, and vertical, which gravity slows, reverses, and speeds back up:

vₓ = v₀ × cos(θ), v_y = v₀ × sin(θ)
  • v₀ — launch speed, in metres per second (m/s)
  • θ (theta) — launch angle above the horizontal
  • vₓ — horizontal velocity, constant through the flight
  • v_y — initial vertical velocity

Because vₓ = v₀ cos(θ) and v_y = v₀ sin(θ), the two components and the launch speed form a right triangle — vₓ² + v_y² = v₀² — the same relationship the Pythagorean Theorem Calculator solves directly.

These two components then feed into the range, peak height and flight time formulas:

Projectile motion formulas for range, peak height and flight time
Solving forFormula
Range (level ground)R = (v₀² × sin(2θ)) ÷ g
Peak heightH = h₀ + (v₀² × sin²(θ)) ÷ (2g)
Flight time (level ground)T = (2 × v₀ × sin(θ)) ÷ g

When the launch height and landing height differ, range and flight time use the general form instead: flight time comes from solving the vertical position equation y(t) = h₀ + v_y t − ½gt² for its positive root, and range is then vₓ × T.

Worked example

A ball launched at 20 m/s at a 45° angle on level ground, under standard Earth gravity (9.81 m/s²):

  • vₓ = 20 × cos(45°) ≈ 14.14 m/s
  • v_y = 20 × sin(45°) ≈ 14.14 m/s
  • Peak height: H = 14.14² ÷ (2 × 9.81) ≈ 10.19 m
  • Flight time: T = (2 × 20 × sin(45°)) ÷ 9.81 ≈ 2.88 s
  • Range: R = (20² × sin(90°)) ÷ 9.81 ≈ 40.77 m

What does this calculator assume?

This calculator models ideal projectile motion: constant gravity and no air resistance, wind or spin. Launch height and landing height are optional and default to 0 (a level-ground launch) — nothing about the flight is inferred beyond what these figures and the formulas above describe. It is an educational physics calculator, not an air-resistance simulator or trajectory-plotting tool.

Want the kinetic energy of the launched object as well? Combine its launch speed with its mass in the Kinetic Energy Calculator.

Frequently asked questions

What is the formula for projectile motion?

Projectile motion splits launch speed v₀ into a horizontal component vₓ = v₀ × cos(θ) and a vertical component v_y = v₀ × sin(θ), where θ is the launch angle. Gravity g only acts on the vertical component, so horizontal velocity stays constant through the flight while vertical velocity slows, stops at the peak, then speeds up again on the way down. Range, peak height and flight time are all worked out from these two components.

How do you calculate the range of a projectile?

On level ground, range is R = (v₀² × sin(2θ)) ÷ g. With a launch height or a different landing height, the calculator first finds the flight time T and then uses R = vₓ × T. For example, a launch speed of 20 m/s at 45° on level ground with standard gravity gives R = (20² × sin(90°)) ÷ 9.81 ≈ 40.77 m.

How do you calculate the peak height of a projectile?

Peak height is H = h₀ + (v₀² × sin²(θ)) ÷ (2g), where h₀ is the launch height. It depends only on the vertical component of the launch speed and gravity — the landing height plays no part, since the projectile reaches its highest point on the way up regardless of where it eventually lands.

How do you calculate the flight time of a projectile?

On level ground, flight time is T = (2 × v₀ × sin(θ)) ÷ g. With a launch height h₀ and landing height h_land that differ, the calculator solves the vertical position equation for the positive root: T = (v_y + √(v_y² + 2g(h₀ − h_land))) ÷ g, where v_y = v₀ × sin(θ).

What happens if the landing height is higher than the projectile can reach?

If the landing height entered is above the maximum height the projectile reaches (its peak height), there is no valid flight time or range — the object never gets that high, so it can never land there. The calculator flags this on the landing height field rather than showing an impossible result. Peak height itself is unaffected, since it does not depend on the landing height.

Does this calculator account for air resistance?

No. This calculator uses ideal projectile motion: constant gravity, no air resistance, and no wind. This is the standard model taught in introductory physics and is exact in a vacuum, but a real object — especially a light or irregularly shaped one — will fall short of these figures because drag slows it down.

What units does this calculator use?

SI units throughout: launch speed in metres per second, angle in degrees, launch and landing height in metres, gravity in metres per second squared, and results in metres and seconds.

What value of gravity should I use?

Standard Earth gravity, 9.81 m/s², is the default and fits almost every everyday example. Use 1.62 m/s² for the Moon, 3.71 m/s² for Mars, or any other value from 0.1 to 30 m/s² to explore a different world or a hypothetical scenario.

Last updated 25 October 2026.