Projectile Motion Calculator: Range, Height and Flight Time

Enter a launch, choose a horizontal scenario, or solve for the speed or angles needed to hit a target. The calculator returns the trajectory, flight quantities, velocities, graph, and worked steps using constant gravity with no air resistance.

Set up the problem

Try a scenario

Solve for

Standard launch
Speed + angle → results

Horizontal launch
Speed + height → results

Required speed
Angle + target → speed

Required angle
Speed + target → angle(s)

Values and units

Unit system

Launch speed

Must be zero or greater.

Speed unit

Launch angle

Measured above horizontal; a negative value aims down.

Angle unit

Initial height

Height above the y = 0 reference level.

Height and distance unit

Position and velocity at time

Seconds after launch.

Gravity

Advanced, copy and sharing

Displayed decimal places

Calculations stay in your browser. Sharing puts only the displayed inputs in the URL.

Press Ctrl/Cmd + Enter to calculate.

Answers

Show calculation steps

Quantitative trajectory graph

The x and y axes are scaled independently so values remain legible; use the numbered axes rather than the visual slope to read the launch angle.

Trajectory data table

Sampled trajectory values for the primary solution
Point	Time (s)	x (m)	y (m)	vₓ (m/s)	vᵧ (m/s)

Projectile motion equations and method

Projectile motion is the two-dimensional motion of an object acted on only by constant downward gravity. Resolve the launch velocity into horizontal and vertical components, then apply constant-acceleration kinematics to each direction independently.

How to use this calculator

Choose the quantity you want to solve.
Select units and enter the known speed, angle, initial height, and—when needed—target coordinates.
Choose Earth, Moon, Mars, or a custom positive gravity magnitude.
Select Calculate. Read answers first, then expand the worked steps and trajectory table.

Variables and governing equations

v₀: launch speed; θ: launch angle; h₀: initial height; g: positive gravity magnitude.
v_x = v₀ cos θ and v_y = v₀ sin θ.
x(t) = v_xt and y(t) = h₀ + v_yt − ½gt².
v_y(t) = v_y − gt; horizontal velocity remains constant.
Trajectory: y(x) = h₀ + x tan θ − gx²/(2v₀² cos²θ).

Level-ground and elevated launches

When launch and landing heights are equal, t = 2v₀sinθ/g, R = v₀²sin(2θ)/g, and the ideal maximum-range angle is 45°. From height h₀, flight time is the positive root of h₀ + v_yt − ½gt² = 0. The optimal angle becomes tan⁻¹[v₀/√(v₀² + 2gh₀)]; for 30 m/s from 2 m on Earth it is about 44.4°, not 41°.

Assumptions and limits

This is an ideal point-mass model. It ceases to closely match reality when air drag, wind, lift or spin matter; when the projectile travels far enough for Earth curvature or changing gravity to matter; or when terrain is not represented by the selected endpoint height. Mass does not enter the ideal equations, but size, shape, and mass affect real drag.

Verified worked examples

1. Level-ground angled launch

Given: v₀ = 20 m/s, θ = 45°, h₀ = 0, g = 9.80665 m/s².

Substitute: v_x = v_y = 20/√2 = 14.1421 m/s. Then t = 2(14.1421)/9.80665.

Answer: flight time 2.884 s, range 40.789 m, maximum height 10.197 m. The equal components produce the familiar symmetric ideal arc.

2. Horizontal launch from a cliff

Given: v₀ = 12 m/s horizontally from 20 m on Earth.

Substitute: 0 = 20 − ½(9.80665)t², so t = √[40/9.80665]. Range = 12t.

Answer: flight time 2.020 s and range 24.236 m. Horizontal speed does not change; falling time is set by height and gravity.

3. Two angles to one target

Given: v₀ = 25 m/s, target (x, y) = (40 m, 5 m), h₀ = 0 on Earth.

Substitute: solve the quadratic in tan θ: tan θ = [v² ± √(v⁴ − g(gx² + 2Δyv²))]/(gx).

Answer: approximately 27.7° (low arc) or 69.4° (high arc). Both hit the coordinate; the high arc takes longer.

Methodology and review

Author: Starlight Robotics Editorial Team

Physics review: Starlight Robotics science review, classical mechanics

Reviewed: 17 July 2026

Gravity standard: 9.80665 m/s², the conventional standard acceleration of free fall (g_n).

Numerical checks cover level, elevated, downward, horizontal, lunar, Martian, reachable-target, and unreachable-target cases.

References: NIST standard acceleration of gravity; OpenStax Physics: Projectile Motion.

No ratings or performance claims are fabricated in the structured data.

Projectile motion FAQs

Why is 45° optimal only when launch and landing heights match?

On level ground, range is proportional to sin(2θ), which peaks at 45°. An elevated launch already has extra falling time, so a slightly lower angle usually travels farther.

Does mass affect ideal projectile motion?

No. Mass cancels when gravity is the only force. In reality, mass can change how strongly drag affects an object of a given size and shape.

How does initial height change the result?

More initial height usually increases flight time and range. Enter the launch height above the y = 0 landing reference; the calculator solves the full quadratic.

How does a horizontal launch work?

Choose Horizontal launch. The initial vertical velocity is zero, while horizontal velocity stays constant and gravity determines the fall.

Why can two target angles exist?

At sufficient speed, a shallow, fast low arc and a steep, slower high arc can cross the same coordinate. If the target-angle discriminant is negative, neither arc is possible at that speed.

Is air resistance included?

No. The model assumes vacuum-like motion with constant gravity. Drag, wind, spin, and lift can materially change a real trajectory.

How do radians and degrees differ?

They measure the same angle on different scales: 180° = π radians. Choose either angle unit; the calculator converts to radians internally for trigonometry.

Which gravity value should I use?

Use 9.80665 m/s² for standard Earth-gravity exercises, a local measured value for experiments, or the Moon/Mars presets for ideal comparisons. Custom gravity uses the selected length unit per second squared.