Find the time of flight, range, maximum height and impact speed of a projectile from its launch speed, angle and height, on Earth, the Moon or Mars.
m/s
°
m
Enter the values to see the result here.
Formula
With launch speed v0, angle θ, height h0 and gravity g, the velocity components are vx=v0cosθ and vy=v0sinθ:
t=gvy+vy2+2gh0R=vxtH=h0+2gvy2
The impact speed is vx2+(vy−gt)2.
Worked example
A ball is kicked at 20 m/s at 45° from the ground on Earth.
vx=vy=20cos45°=14.14 m/s.
Time of flight: t=2×14.14/9.807=2.88 s.
Range: 14.14×2.88=40.8 m.
Maximum height: 14.142/(2×9.807)=10.2 m.
It lands at the same 20 m/s, 45° below the horizontal.
Common mistakes
Entering the angle in radians. This tool expects degrees.
Measuring the angle from the vertical instead of from the horizontal.
Forgetting the launch height, which can add a lot of range for a throw from a tall building.
Using the result for light objects such as a shuttlecock, where air drag dominates.
Frequently asked questions
Which launch angle gives the longest range?
From level ground, 45 degrees. When you launch from a height above the landing level, the best angle is a little less than 45 degrees. Complementary angles such as 30 and 60 give the same range from level ground.
Does mass change a projectile's path?
Not without air resistance: every mass follows the same path, because gravity accelerates all objects equally. With air resistance, a light object loses more speed than a dense one.
Why are real ranges shorter than the calculated ones?
Drag opposes motion and grows with speed, so a fast ball, arrow or bullet falls well short of the vacuum value. This calculator is accurate for slow, dense objects and for teaching.