Projectile Motion Calculator

How to use this tool

Choose what to work out. “Range and flight of a throw” takes a launch speed, angle and height and gives the range, flight time, highest point, and the speed and angle at impact, with the path drawn to scale. “Angle to hit a target” returns the low and high launch angles for a distance and height, or says the target is out of reach and how far that speed can go. “Throw with air drag” adds a drag coefficient, mass, size and air density, and draws the vacuum and drag paths together.

The input people most often get wrong is the angle. It is measured up from the horizontal, so 45 is a diagonal and 90 is straight up. Speed and distance units are separate choices, and 60 typed with m/s selected is not 60 mph. Launch height is measured above the landing level, so a ball released 2 m above the ground is 2. For a target, height is measured from the launch point, so the ground seen from a 2 m platform is −2.

What the calculation is doing

With no air the two directions separate. Horizontally nothing acts on the object, so the speed stays vx = v cos θ. Vertically gravity pulls at g, so vy = v sin θ − g t. Setting the height to zero gives the time of flight T = (vy + √(vy² + 2 g h)) ÷ g, where h is the launch height, and the range is vx × T. The highest point is h + vy² ÷ (2 g). The impact speed is √(v² + 2 g h), which is conservation of energy: a fall of h adds 2 g h to the speed squared. Gravity is the standard 9.80665 m/s², NASA’s values for the Moon (1.62), Mars (3.73) and Jupiter (25.92 at the 1 bar level), or your own.

Aiming at a target a horizontal distance x away and a height y above the launch point turns the path equation into a quadratic in tan θ: tan θ = (v² ± √(v⁴ − g (g x² + 2 y v²))) ÷ (g x). The two roots are the low and high angles. A negative number under the root means the target lies outside the envelope of everything that speed can reach, and the slowest launch that gets there is √(g (y + √(x² + y²))).

Drag has no closed form. With quadratic drag the acceleration is a = −(½ ρ Cd A ÷ m) |v| v − g, where ρ is the air density, Cd the drag coefficient, A the frontal area and m the mass, the drag equation NASA Glenn publishes. The tool steps this forward with fourth order Runge Kutta, halves the step until the range agrees to one part in 10¹⁰, and shortens the last step to land exactly on the ground. The best angle with drag comes from a golden section search.

Why 45 degrees is not the answer to everything

The rule that 45° throws farthest is true for one case: launch and landing at the same height, and no air. The range there is v² sin 2θ ÷ g, and sin 2θ peaks at 45°. It also explains complementary angles. Since sin 2θ is the same for θ and 90° − θ, 30° and 60° at 30 m/s both land 79.5 m away, although the steep throw hangs for 5.3 s against 3.06 s. Range also scales with the square of the speed: 20 m/s reaches 40.8 m from level ground and 40 m/s reaches 163.2 m, four times as far.

Launch from above the landing level and the best angle falls. Setting the derivative of the range to zero gives θ = atan(v ÷ √(v² + 2 g h)). At 20 m/s from 2 m that is 43.66°, and the gain over 45° is only 4 cm, because the curve is flat near its top. From a 20 m cliff at 25 m/s the best angle is 38.1° and it reaches 81.3 m against 79.7 m at 45°. Linthorne, who studied throwing events, gives an optimum of 42° for a shot put released at 2.1 m at 13 m/s, and the formula gives 41.9°. Athletes also cannot push as hard at steep angles, which lowers real optimum angles further.

Drag lowers the best angle a second time and takes a large share of the range. A golf ball at 70 m/s with Cd 0.25 and no spin has a vacuum range of 500 m at 45°. With drag its best angle is 38.1° and it reaches only 202 m. A real drive does better because backspin lifts the ball, and this tool has no lift: a COMSOL simulation with spin carries a 7 iron shot at 187 km/h and 17.4° about 150 m, where the no spin model here gives 107 m. Heavy objects hardly notice the air: Linthorne reports almost no aerodynamic effect on the shot put optimum.

Height also decides how steeply the object lands. The default launch at 45° from 2 m arrives at 47.6° below horizontal, steeper than it left. A stone thrown flat at 10 m/s off a 20 m cliff lands 20.2 m out at 22.2 m/s and 63.2° below horizontal, and more height brings the impact closer to vertical.

Best angle by launch height, and the same throw on other worlds

The best angle from the formula, at three launch speeds, for a range of launch heights. Slow throws from high places lose the most: at 10 m/s from 20 m the best angle is barely more than half of 45°.

Now one 20 m/s throw at 45° from level ground under each gravity in the tool. Gravity divides straight into the range and the flight time, so the Moon gives about six times the Earth distance.

A worked example you can check on paper

Default values: 20 m/s at 45° from 2 m on Earth. vx = vy = 20 × 0.7071 = 14.142 m/s. Time to the highest point = 14.142 ÷ 9.80665 = 1.442 s. Highest point = 2 + 14.142² ÷ (2 × 9.80665) = 2 + 10.197 = 12.2 m. Time of flight = (14.142 + √(200 + 39.227)) ÷ 9.80665 = (14.142 + 15.467) ÷ 9.80665 = 3.019 s. Range = 14.142 × 3.019 = 42.7 m. Impact speed = √(400 + 39.227) = 20.96 m/s, and the impact angle is atan(15.467 ÷ 14.142) = 47.6° below horizontal.

Aiming: a level target 50 m away at 30 m/s. v⁴ = 810,000 and g x² = 9.80665 × 2,500 = 24,516.6, so the root is √(810,000 − 9.80665 × 24,516.6) = √569,574 = 754.7. Then tan θ = (900 ∓ 754.7) ÷ (9.80665 × 50) = 0.296 or 3.375, which are 16.5° and 73.5°.

Mistakes that catch people out

Measuring the angle from the vertical. A launch angle of 30 here is 30 degrees above the ground. If your figure is measured from the vertical, enter 90 minus that.

Mixing units. Speed and distance units are chosen separately. A pitch of 90 with m/s selected is 324 km/h, and the range will be absurd.

Getting the sign of the target height wrong. Target height is measured from the launch point. Ground below a 2 m platform is −2, not 2, and the low angle can come out negative, meaning a throw aimed slightly downward.

Assuming 45 degrees is always best. It is best only on level ground with no air. From a height, or with drag, aim lower.

Trusting the vacuum answer for a light, fast object. A baseball thrown at 40 m/s loses 45% of its vacuum range to air. Use the drag mode, which still leaves out spin and wind.

Popular calculations

Each opens the calculator with those figures filled in, and they can be changed once it is open. The drag examples use a drag coefficient of 0.47 for a smooth sphere and 0.25 for a dimpled golf ball, unless stated. Golf balls fly farther in real life because spin adds lift that this tool leaves out.

Throws, kicks and drops on Earth

Moon, Mars and Jupiter

Find the angle to hit a target

Air drag against a vacuum

How this was checked

The closed form vacuum results were compared with a separate Dormand Prince 5(4) integrator (tolerance 10⁻¹³) over 4,847 seeded random launches, and with SciPy solve_ivp (DOP853) over 2,500 of them. The largest relative difference in time of flight, range, maximum height, time to apex, impact speed and impact angle was 5.7 × 10⁻¹¹ against the first and 5.7 × 10⁻¹¹ against SciPy. Drag mode was compared the same way over 2,860 random cases, with Cd from 0.05 to 1.5, masses from 5 g to 50 kg and air densities from 0.01 to 1.3 kg/m³: the largest difference was 3 × 10⁻¹¹ against the integrator and 3 × 10⁻¹¹ against SciPy, where the target was 10⁻⁶. The angle solver was checked by shooting back through 4,000 random targets.

Sanity limits held to rounding error: complementary angles give equal range, 45° gives v² ÷ g on level ground, doubling the speed quadruples the range, and zero drag reproduces the closed form. Drag mode also matched the exact solutions for a vertical throw and for a drop from rest to about 10⁻⁸. Published examples: OpenStax University Physics Volume 1 section 4.3 gives a fireworks shell at 70.0 m/s and 75.0° reaching 233 m in 6.90 s, and the tool gives 233.25 m and 6.899 s with the book’s g of 9.80. The tennis example gives 3.79 s and 26.5 m/s, and the tool 3.791 s and 26.53 m/s. OpenStax College Physics (chapter 3, projectile motion) has a rock ejected at 25.0 m/s and 35.0° landing 20.0 m lower in 3.96 s at 31.9 m/s and 50.1° below horizontal, and the tool gives 3.958 s, 31.89 m/s and 50.05°. The gaps come from the books rounding intermediate values; with the standard 9.80665 the fireworks time is 6.895 s, 0.005 s from the printed 6.90 s, all from the different g. Linthorne’s 42° shot put optimum is 41.88° here, and HyperPhysics states that complementary launches share a range.

Drag mode is only as good as the coefficient. Raising Cd by 10% on the 40 m/s baseball moves its range by −3.9%, and lowering it by 10% moves it by 4.3%. NASA gives 0.07 to 0.5 for a sphere depending on Reynolds number, so a real ball can sit well away from 0.47. The 1.225 kg/m³ air density is the International Standard Atmosphere sea level value.

Common questions

What launch angle gives the maximum range?

On level ground with no air, 45 degrees, because the range is v² sin 2θ ÷ g and sin 2θ peaks there. Launch from above the landing level and the best angle is atan(v ÷ √(v² + 2 g h)), always below 45 degrees: 43.7° for 20 m/s from 2 m, 38.1° for 25 m/s from a 20 m cliff. With air drag it is lower again.

How do you calculate the range of a projectile?

Split the launch speed into vx = v cos θ and vy = v sin θ. The time of flight is (vy + √(vy² + 2 g h)) ÷ g, and the range is vx times that time. For 20 m/s at 45° from 2 m that is (14.14 + 15.47) ÷ 9.80665 = 3.019 s and 14.14 × 3.019 = 42.7 m.

Why do two different angles give the same range?

Because sin 2θ is the same for θ and 90° − θ. From level ground at 30 m/s, 30° and 60° both land 79.5 m away. The steep throw is in the air 5.3 s against 3.06 s, and it climbs 34.4 m against 11.5 m. Launch from a height and the pair is no longer exactly equal, and drag breaks the symmetry too.

How far will a ball go?

In a vacuum a 40 m/s throw at 35° from 2 m goes 156.1 m. A baseball with Cd 0.47 in sea level air goes 85.6 m, a loss of 45%. Spin and a changing drag coefficient make the drag figure an estimate, not a measurement.

How much does air resistance change a throw?

It depends on speed, mass and size, because drag grows with the square of speed and is divided by the mass. A 145 g baseball at 40 m/s loses 45% of its vacuum range, a 5 kg ball at 100 m/s far less in proportion, and a 20 g ball of the same size most of it. Drag also steepens the landing and lowers the best angle.

Does the calculator include spin, wind or the curve of the Earth?

No. There is no spin or lift, no wind, a constant drag coefficient, constant air density and a flat Earth with constant gravity. Those are fine for throws of tens to hundreds of metres. Beyond about 20 km of range or height the tool warns you.

Related Calculators and Reference

• Kinetic Energy Calculator: the energy of a projectile at launch and impact, from the speeds this tool gives.
• Vector Calculator: add, resolve and rotate the velocity components used here.
• Trig Calculator: the sines, cosines and inverse tangents behind every angle in the working.
• Density Altitude Calculator: air density at your elevation, to enter in the drag mode.
• Speed Converter: Kmh to Mph: convert a radar gun or speedometer reading before you enter it.
• Stopping Distance Calculator: the other place constant deceleration and speed squared meet, in braking.
• OpenStax University Physics Volume 1, Projectile Motion: the textbook derivation and worked examples used to check this tool.