Projectile Motion Calculator

A projectile motion calculator finds where and when a launched object lands using the standard kinematic equations with gravity g = 9.81 m/s². Launch at 20 m/s and 45° from ground level and it flies for 2.88 seconds, peaks at 10.2 m, and lands 40.8 m away. Enter launch speed, angle, and starting height for instant range, max height, time of flight, and impact velocity.

Projectile motion is one of several math tools here. For non-physics arithmetic, reach instead for the Percentage Calculator (what is X% of Y), the Percentage Increase Calculator and Percentage Change Calculator for growth between two numbers, or the Discount Calculator for sale prices - use this projectile tool only when speed, angle, and gravity are involved.

The core idea: split the motion into two directions

Projectile motion looks complicated because the object follows a curved path (a parabola), but the math is simple once you separate it into two independent one-dimensional problems. The launch velocity v at angle θ splits into a horizontal component and a vertical component:

The horizontal velocity v_x never changes - there is no horizontal force in the ideal model, so the object drifts sideways at a steady rate. The vertical velocity v_y is the only part gravity touches: it slows the object on the way up, reaches zero at the top, and speeds it back up on the way down. Treating the two directions separately is the whole trick to projectile problems.

The formulas this calculator uses

For a projectile launched from height h with speed v at angle θ, under gravity g = 9.81 m/s²:

That t is the total time of flight - the positive solution of the vertical position equation y(t) = h + (v·sinθ)t − ½gt² = 0. The remaining results follow directly:

• Maximum height: H = h + (v·sinθ)² ÷ (2g)
• Horizontal range: R = v_x × t = (v·cosθ) × t
• Time to apex: t_up = (v·sinθ) ÷ g
• Impact speed: √(v_x² + v_y,impact²), where v_y,impact = v·sinθ − g·t

When the launch and landing heights are equal (h = 0), the range formula simplifies to the famous R = v² × sin(2θ) ÷ g, and time of flight becomes t = 2v·sinθ ÷ g.

A full worked example

Take the default inputs: v = 20 m/s, θ = 45°, h = 0. First, split the velocity. Since cos 45° = sin 45° ≈ 0.7071, both components are v_x = v_y = 20 × 0.7071 ≈ 14.14 m/s.

Next, the time of flight. With h = 0 the formula collapses to t = 2 × 14.14 ÷ 9.81 ≈ 2.88 s. The apex arrives at half of that, about 1.44 s. Maximum height is H = 14.14² ÷ (2 × 9.81) = 200 ÷ 19.62 ≈ 10.19 m. The range is R = 14.14 × 2.88 ≈ 40.77 m. Because the object lands at the same height it left, the impact speed equals the launch speed, 20 m/s, but now angled downward at 45°. Every number the calculator shows traces back to these few lines of algebra.

How to use this projectile motion calculator

1. Launch speed: enter the initial speed in meters per second. If your value is in mph, multiply by 0.447; if it is in feet per second, multiply by 0.3048.
2. Launch angle: type the angle above the horizontal in degrees, or tap one of the quick presets (15°, 30°, 45°, 60°, 75°) to compare.
3. Initial height: leave it at 0 for a ground launch, or enter the height in meters if the object starts on a cliff, table, or platform.
4. Read the results: the headline number is the horizontal range, with time of flight, max height, time to apex, and impact speed below it, plus a full velocity-component breakdown.

Change any input and recalculate to see instantly how steeper angles trade range for height, or how a higher launch point stretches the distance.

Convert mph and feet to the metric inputs

The model is metric, so convert imperial values before entering them: multiply mph by 0.44704 for m/s, ft/s by 0.3048 for m/s, and feet by 0.3048 for meters. Common conversions:

Exact factors: 1 mph = 0.44704 m/s, 1 ft/s = 0.3048 m/s, 1 ft = 0.3048 m. The ft/s and feet columns share the 0.3048 factor.

Who this calculator is for

• Physics students checking homework on kinematics and 2D motion.
• Teachers and lab instructors generating expected values for projectile experiments.
• Engineering and game-dev hobbyists sanity-checking trajectory math before coding it.
• Sports and ballistics enthusiasts curious how launch angle and speed set the distance of a throw, kick, or shot.
• Anyone who wants a quick, exact answer to "how far and how high will this go?"

A second example: launched from a height

Suppose you throw a ball at 15 m/s, 30°, off a 10 m cliff. The vertical component is v_y = 15 × sin 30° = 7.5 m/s and the horizontal is v_x = 15 × cos 30° ≈ 12.99 m/s. The discriminant is 7.5² + 2 × 9.81 × 10 = 56.25 + 196.2 = 252.45, whose square root is about 15.89. Time of flight is t = (7.5 + 15.89) ÷ 9.81 ≈ 2.38 s - longer than a ground launch because the ball has farther to fall. The range is 12.99 × 2.38 ≈ 30.97 m, and the max height is 10 + 7.5² ÷ (2 × 9.81) ≈ 12.87 m. The impact speed climbs above the 15 m/s launch speed because the ball converts the extra 10 m of height into additional kinetic energy on the way down.

Projectile motion example solved step by step

Here is that same launch (v = 15 m/s, θ = 30°, h = 10 m) broken into the exact stages the calculator runs, so you can follow every number:

Values computed with g = 9.81 m/s², no air resistance. The vertical impact velocity is v_y − g·t = 7.50 − 9.81 × 2.38 = −15.89 m/s (downward).

How launch angle changes the result

Holding speed at 20 m/s on flat ground, watch how the angle reshapes the trajectory:

• 15°: a flat, fast shot - short max height (~1.4 m) but only ~20.4 m of range.
• 30°: range ~35.3 m, max height ~5.1 m.
• 45°: the range peak at ~40.8 m, max height ~10.2 m.
• 60°: range falls back to ~35.3 m (the same as 30°) but max height jumps to ~15.3 m.
• 75°: a steep lob - ~20.4 m of range but ~19 m of height.

Notice that complementary angles (30° and 60°, or 15° and 75°) give the same range on level ground - one is a flat liner, the other a high arc. That symmetry comes straight from sin(2θ) in the range formula.

Projectile range at 30°, 45° and 60° by launch speed

This matrix shows range, maximum height, and time of flight for common launch speeds at the three most-asked angles, all on level ground (h = 0). Range peaks at 45° for every speed, and doubling the speed roughly quadruples the range.

All values computed with the calculator's own formulas at g = 9.81 m/s², ground launch (h = 0), no air resistance. The 45° row (highlighted) is the range maximum at every speed.

Key terms explained

• Projectile: any object moving under gravity alone after launch, with no propulsion.
• Trajectory: the curved path the object traces - a parabola in the ideal model.
• Time of flight: how long the object stays airborne, from launch to landing.
• Apex: the highest point of the path, where vertical velocity is momentarily zero.
• Range: the horizontal distance covered from launch to landing.
• Components: the horizontal (v_x) and vertical (v_y) pieces the launch velocity is split into.
• g: gravitational acceleration near Earth's surface, taken as 9.81 m/s² here.

What changes the result the most

• Launch speed: the strongest lever. Range scales with the square of speed, so doubling the speed quadruples the distance on flat ground.
• Launch angle: sets the balance between height and range; 45° maximizes distance on level ground.
• Initial height: a higher launch point adds flight time and stretches the range, and lowers the optimal angle below 45°.
• Gravity: a smaller g (say, on the Moon) would dramatically increase both range and flight time; this calculator fixes g at Earth's 9.81 m/s².

Tips for getting accurate answers

• Convert units first. The model is metric - turn mph, ft/s, and feet into m/s and meters before entering them.
• Measure the angle from horizontal, not from vertical. A 0° launch is perfectly flat; 90° goes straight up.
• Check the symmetry. For a ground launch, if your impact speed does not equal the launch speed, you likely entered a non-zero height by mistake.
• Use complementary angles to confirm a range result - 40° and 50° should match on flat ground.

Limitations and assumptions

This is the idealized textbook model, which is exact in a vacuum but an approximation in real air. Keep these in mind:

• It ignores air resistance (drag), which shortens real ranges - sometimes dramatically for fast or light objects like a tennis ball or bullet.
• It ignores wind and spin (the Magnus effect that curves a soccer or golf ball).
• It treats g as constant at 9.81 m/s²; g actually varies slightly with altitude and latitude (about 9.78 to 9.83 m/s²).
• It assumes a flat Earth and level ground apart from the initial-height offset, so it is not meant for very long-range ballistics.
• It models a point mass, so object size, shape, and the launch point's exact geometry are not considered.

About this formula

The equations used here are the standard one-dimensional kinematic equations of constant acceleration, applied independently to the horizontal and vertical directions. They are a deterministic mathematical standard taught in every introductory physics course and require no external data source - given the same inputs, the calculator returns the same exact values every time. The only physical constant chosen is g = 9.81 m/s², the conventional average gravitational acceleration at Earth's surface.

Related calculators

• For right-triangle and component work behind the velocity split, see the Pythagorean Theorem Calculator and the Trigonometry Calculator.
• To solve the underlying quadratic for time of flight by hand, use the Quadratic Formula Calculator.
• To convert mph or feet into the metric inputs, use the Unit Converter or the Feet to Meters Converter.