Use this free projectile motion calculator to find the range, maximum height, time of flight and final velocity of a projectile from its launch speed, angle and starting height. Choose m/s, km/h, mph or ft/s, meters or feet, and the gravity of Earth, the Moon, Mars, Jupiter or your own. The At a Given Time tab gives the position, height and speed at any moment, and the Hit a Target tab finds the launch angle (low and high arc) or the speed needed to reach a target. The path is drawn with the peak, the landing, H, R and the launch angle, and the working is shown under every answer. Tables list the projectile motion formulas, the range at different angles and the same throw on other worlds. It is free and instant, and your numbers stay in your browser.
What is projectile motion?
A projectile is anything thrown, kicked or fired that then moves only under gravity: a football, a stone, a water jet or a ball rolling off a table. It moves sideways at a steady speed while gravity pulls it down, and the two motions do not affect each other. Put together, they make a curved path called a parabola. The calculator finds the range, the maximum height and the time of flight, and draws the path.
The projectile motion formulas
Split the launch speed v at the angle θ into a horizontal part, vₓ = v × cos θ, and a vertical part, vᵧ = v × sin θ. Gravity g, which is 9.81 m/s² on Earth, only changes the vertical part.
Time to the peak = vᵧ ÷ gMaximum height = h₀ + vᵧ² ÷ (2 × g)Time of flight = (vᵧ + √(vᵧ² + 2 × g × h₀)) ÷ gRange = vₓ × time of flight, or v² × sin(2θ) ÷ g from level groundWorked example: 20 m/s at 45°
With v = 20 m/s and θ = 45°, vₓ = vᵧ = 14.14 m/s. The time to the peak is 14.14 ÷ 9.81 = 1.44 s, so the time of flight is 2.88 s. The range is 14.14 × 2.88 = 40.79 m, and the maximum height is 14.14² ÷ (2 × 9.81) = 10.2 m. It lands at the same speed it left, 20 m/s, at 45° below the horizontal.
Launching from a height
If the object starts above the ground, such as a ball thrown from a cliff, put the height in as h₀. The flight is longer because it has further to fall. A ball thrown flat (0°) at 15 m/s from 10 m up stays in the air for √(2 × 10 ÷ 9.81) = 1.43 s and lands 21.42 m away.
The −16 formula in feet
In feet, g is about 32 ft/s², so half of g is about 16, and the height at time t is y = −16 × t² + vᵧ × t + h₀. For a ball thrown straight up at 48 ft/s from 6 ft, the height is 42 ft after 1.5 s, which is the peak, and it lands after 3.12 s. Choose feet and ft/s in the calculator and it does this for you.
Which angle goes the furthest?
From level ground, 45° gives the longest range. Two angles that add up to 90° land in the same place: 30° and 60° both go 35.32 m at 20 m/s, and so do 15° and 75° at 20.39 m. The steeper angle climbs higher and stays up longer. When you launch from a height, the best angle is a little below 45°.
Hitting a target
To hit a target a given distance away, there are usually two launch angles: a low, flat arc and a high, lobbed one. At 20 m/s a target 30 m away on level ground can be hit at 23.67° or 66.33°. If the speed is too low the target is out of reach and there is no angle. The Hit a Target tab also finds the speed you need for an angle you choose.
Gravity on other worlds
Change the gravity to see the same throw on the Moon (1.62 m/s²), Mars (3.71 m/s²) or Jupiter (24.79 m/s²). On the Moon the 20 m/s throw at 45° goes 246.9 m, about six times as far as on Earth, and on Jupiter only 16.14 m.
What this calculator leaves out
It assumes no air resistance, no wind and no spin, and a flat, level landing place. That is a good model for a heavy, compact object at low speed, and the standard model for a school physics problem. Air resistance makes real objects land shorter and lower, especially fast or light ones. For related sums, see the Scientific Calculator for sines and square roots, or the Ohm's Law Calculator for another physics formula.
