Projectile Motion Calculator

Calculate projectile range, maximum height, and flight time from initial velocity and launch angle. Assumes flat ground and no air resistance.

Examples

Optimal angle (20 m/s at 45°)

Range = 40.77 m, Max height = 10.19 m

Initial Velocity: 20 m/s
Launch Angle: 45 °

Range

40.7747 m

Maximum Height

10.1937 m

Total Flight Time

2.8832 s

Time to Peak

1.4416 s

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How It Works

Formula

R = \frac{v^2 \sin(2\theta)}{g}

H = \frac{v^2 \sin^2(\theta)}{2g}

T = \frac{2 v \sin(\theta)}{g}

t_{peak} = \frac{v \sin(\theta)}{g}

Variables, symbols and units

$R$: Horizontal range(m)
$H$: Maximum height(m)
$T$: Total flight time(s)
$t_{peak}$: Time to reach peak(s)
$v$: Initial velocity(m/s)
$\theta$: Launch angle from horizontal(°)
$g$: Gravitational acceleration (9.81)(m/s²)

Calculation method explained

Enter the initial velocity in m/s and the launch angle in degrees. The calculator computes the horizontal range, maximum height, total flight time, and time to reach peak height using standard projectile motion equations with g = 9.81 m/s².

Decompose the initial velocity into horizontal (v·cos θ) and vertical (v·sin θ) components. Gravity acts only on the vertical component, which traces a symmetric up-down parabola. The calculator solves the resulting kinematic equations in closed form: range = v²·sin(2θ)/g, peak height = v²·sin²(θ)/(2g), time to peak = v·sin(θ)/g, total flight = twice the time to peak. Air resistance is neglected.

References and source material

OpenStax University Physics Volume 1 - 4.3 Projectile Motion

Examples

Optimal angle (20 m/s at 45°)20 m/s · 45 ° → 40.7747 m

Range = 40.77 m, Max height = 10.19 m

Initial Velocity: 20 m/s
Launch Angle: 45 °

Range: 40.7747 m

Low angle (30 m/s at 20°)30 m/s · 20 ° → 58.9713 m

Range = 60.17 m, Max height = 5.36 m

Initial Velocity: 30 m/s
Launch Angle: 20 °

Range: 58.9713 m

High angle (15 m/s at 70°)15 m/s · 70 ° → 14.7428 m

Range = 14.76 m, Max height = 10.13 m

Initial Velocity: 15 m/s
Launch Angle: 70 °

Range: 14.7428 m

Frequently Asked Questions

What is projectile motion?

Projectile motion is the motion of an object thrown or projected into the air, subject only to gravitational acceleration. The path forms a parabola.

What angle gives maximum range?

45 degrees gives the maximum range for a given initial speed on flat ground with no air resistance. Complementary angles (e.g., 30° and 60°) give the same range.

Does air resistance affect the results?

This calculator assumes no air resistance (vacuum conditions). In reality, drag significantly reduces range and max height, especially at high speeds.

What value of g is used?

The calculator uses g = 9.81 m/s², the standard gravitational acceleration at Earth's surface.

What are the key formulas?

Range: R = v²sin(2θ)/g. Max height: H = v²sin²(θ)/(2g). Flight time: T = 2v·sin(θ)/g. Time to peak: t = v·sin(θ)/g.

Projectile Motion Calculator

Optimal angle (20 m/s at 45°)

How It Works

Examples

Frequently Asked Questions

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