Projectile Motion Calculator
Range, flight time, peak height and landing speed of anything thrown or launched, with its path drawn, from speed, angle and height.
Result…
How it's calculated
vx = v₀ cos θ, vy = v₀ sin θ Flight time t = (vy + √(vy² + 2gh)) ÷ g Range = vx × t Peak height = h + vy² ÷ 2g Impact speed = √(vx² + vy² + 2gh)
Without air resistance the horizontal speed never changes and gravity pulls the vertical speed down by g every second, so the path is a parabola. The flight time comes from solving h + vy·t − ½gt² = 0 for the positive root; multiply it by the horizontal speed for the range. From flat ground the longest range is at 45°, but from a height a lower angle goes further. Gravity defaults to standard gravity, 9.80665 m/s² (32.17405 ft/s²), and can be changed for other places or planets. Imperial inputs are converted with 1 ft = 0.3048 m and 1 mile = 1,609.344 m exactly.
Your numbers, step by step
About the Projectile Motion calculator
This projectile motion calculator follows anything thrown, kicked or launched: from the launch speed, the angle above the horizontal and the launch height, it gives the range, the flight time, the highest point, the time to reach it and the speed and angle at which it lands. It draws the path as a curve of height against distance. Speeds can be entered in m/s, km/h, ft/s or mph, and distances come out in metres or feet. Gravity starts at standard gravity, 9.80665 m/s² (32.17405 ft/s²), and can be changed for the Moon or another planet. Physics students use it for kinematics problems, and it answers questions about a ball or a water jet. The model leaves out air resistance, so it gives the ideal path. The steps show each formula with your numbers.
How to use it
- Pick “Metric” or “Imperial” under “Units”, and the “Speed unit”.
- Enter the “Launch speed” and the “Launch angle” in degrees (negative throws downward).
- Enter the “Launch height” above the ground where it lands, or 0 for flat ground.
- Leave “Gravity” at standard gravity or change it.
- Read the range, flight time, peak height and landing speed, and look at the drawn path.
Worked example
A ball thrown at 15 m/s and 35° from 1.5 m high has a horizontal speed of 12.29 m/s. It rises for 0.877 s to a peak of 5.27 m, stays in the air for 1.914 s and lands 12.29 × 1.914 = 23.52 m away, moving at 15.95 m/s at 39.6° below the horizontal.
Updated
Questions people ask
How do I calculate the range of a projectile?
Split the launch speed into horizontal (v cos θ) and vertical (v sin θ) parts, find the flight time t = (vy + √(vy² + 2gh)) ÷ g, then multiply the horizontal speed by that time. On flat ground this becomes v² sin 2θ ÷ g.
What launch angle gives the longest range?
45° on flat ground with no air resistance. Launched from a height, a lower angle goes further, and with air drag the best angle for a real ball is lower still.
How high does a projectile go?
The peak is the launch height plus vy² ÷ 2g, where vy is the upward part of the speed. The velocity calculator covers the speed at any moment on the way.
Why does my real ball land short of the answer?
Air resistance slows it the whole way, and spin can lift or drop it. The calculator ignores both, so it is the upper limit for a light or fast object.
Can it work out a ball thrown straight up?
Yes: set the angle to 90°. The range is zero, the flight time is the time up and back down, and the path is drawn against time instead of distance.