Projectile Motion Calculator (Range, Maximum Height and Time of Flight)

Use this projectile motion calculator to find the time of flight, the maximum height, the range and the speed at which the object hits the ground. Enter the initial speed, the launch angle and the launch height, or change the gravity to launch on the Moon or Mars. The path is drawn to scale and the time control shows the velocity components at any instant.

Air resistance is not included: this is the ideal model taught in physics courses, where the only acceleration is gravity. Below you will find the formulas, worked examples, tables and frequently asked questions.

How to use the projectile motion calculator

Step 1: Enter the initial speed in m/s, the launch angle in degrees (0° is horizontal, 90° is straight up) and the height the object starts from. You can also move the sliders.

Step 2: Pick the gravity: Earth, Moon, Mars or your own value in m/s².

Step 3: Click Calculate or press Enter. Results also update as you type.

Step 4: Watch the animation. The Time control stops the object anywhere along the path and shows its velocity components. The Calculation box lists every operation with your numbers.

Projectile motion formulas

The motion splits in two: horizontal at constant speed and vertical with acceleration g downward. First break the initial speed v0 into components using the angle θ:

vx = v0 × cos(θ)     vy0 = v0 × sin(θ)

With launch height h, the position at time t is x = vx × t and y = h + vy0 × t - g × t² ÷ 2. The object lands when y = 0, which gives the time of flight:

T = (vy0 + √(vy0² + 2 × g × h)) ÷ g

The maximum height is reached when the vertical velocity is zero, at t = vy0 ÷ g, and the range is the horizontal distance covered during the whole flight:

H = h + vy0² ÷ (2 × g)     R = vx × T

When the launch is from the ground (h = 0), the range simplifies to R = v0² × sin(2θ) ÷ g, and the maximum comes at 45°. At impact the vertical velocity is vy = vy0 - g × T and the total speed is v = √(vx² + vy²).

Assumptions: gravity is constant, there is no air or wind, and the ground is flat at height 0. With air, the real range is shorter, especially at high speeds.

Projectile motion examples

Example 1: A ball thrown at 20 m/s and 45°

A ball leaves the ground at 20 m/s at an angle of 45° on Earth (g = 9.81 m/s²). The components are vx = vy0 = 14.14 m/s. Then:

T = (14.14 + √(200 + 2 × 9.81 × 0)) ÷ 9.81 ≈ 2.883 s

H = 0 + 200 ÷ 19.62 ≈ 10.19 m

R = 14.14 × 2.883 ≈ 40.77 m

The ball stays in the air for 2.883 s, rises to 10.19 m and lands 40.77 m from where it left.

Example 2: A horizontal throw from a cliff

A stone leaves horizontally (angle 0°) at 15 m/s from a 10 m cliff. Since vy0 = 0, the time comes only from the fall:

T = (0 + √(0 + 2 × 9.81 × 10)) ÷ 9.81 ≈ 1.428 s

R = 15 × 1.428 ≈ 21.42 m

The stone hits the water 21.42 m from the edge. Its vertical velocity at impact is -14.01 m/s, the total speed is 20.52 m/s and it enters at 43.04° below the horizontal.

Example 3: The same throw on the Moon

Same speed and angle as example 1, but with g = 1.62 m/s²:

T = (14.14 + √(200 + 2 × 1.62 × 0)) ÷ 1.62 ≈ 17.46 s

H = 0 + 200 ÷ 3.24 ≈ 61.73 m

R = 14.14 × 17.46 ≈ 246.91 m

The range is 6.06 times the Earth range, because the range is inversely proportional to g.

Example 4: Which angle gives the longest range

With v0 = 30 m/s from the ground, the range is R = v0² × sin(2θ) ÷ g. At 45°, sin(90°) = 1:

R = 30² ÷ 9.81 ≈ 91.74 m

At 30° and at 60° the range is the same, 79.45 m, because sin(60°) = sin(120°). Angles that add up to 90° (complementary) have equal range, but at 60° the object climbs higher (34.4 m against 11.47 m) and stays in the air longer.

Range table by launch angle

Data: v0 = 20 m/s, launched from the ground on Earth (g = 9.81 m/s²).

The same launch on different worlds

Data: v0 = 20 m/s, angle of 45°, from the ground.

Frequently asked questions

What is the formula for the range of a projectile?

From the ground, R = v0² × sin(2θ) ÷ g. With v0 = 20 m/s and θ = 45° on Earth: R = 20² × 1 ÷ 9.81 = 40.77 m. If the launch is from a height, find the time of flight first and then R = vx × T.

Which angle gives the maximum range?

45° when the object leaves and lands at the same height. If it is launched from a point higher than the ground where it lands, the best angle is below 45°.

Does the mass of the object change the result?

No. Without air resistance the acceleration of a falling object does not depend on mass, so a tennis ball and a bowling ball launched the same way follow the same path.

How do you find the time of flight when launching from a height?

Solve y = 0 in y = h + vy0 × t - g × t² ÷ 2. The positive solution is T = (vy0 + √(vy0² + 2 × g × h)) ÷ g, which is the one the calculator uses.

What does this calculator ignore?

Air resistance, wind, spin and the curvature of the Earth. In real throws, air shortens the range and makes the way down steeper than the way up.

Learn more

• Projectile motion in physics: examples with answers
• Range of a projectile: formula and examples
• Acceleration due to gravity in physics
• Vertical projectile motion: formulas and examples

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