How to use it
- Type the Launch speed v0 and the Launch angle θ, for example 20 m/s and 45°. Leave Launch height h0 at 0 for a launch from the ground.
- For a landing at a different level, type it in Landing height h1. Leave it empty to land on the ground.
- Read Range (horizontal distance) and Time of flight, then Speed at impact and Angle at impact.
- Look at the Trajectory chart and the Position and speed during the flight table to follow the path, and open How it is worked out for each formula.
The equations
A projectile moves with a constant horizontal speed and a vertical speed that falls by g every second, where g = 9.80665 m/s² is standard gravity. The launch speed v0 is split into vx = v0 cos θ and vy = v0 sin θ. Eliminating the time gives the parabola that the path follows.
vx = v0 cos θ
vy = v0 sin θ − gt
x = vx t
y = h0 + vy t − ½gt²
t = (vy + √(vy² + 2g(h0 − h1))) ÷ g- The range is R = vx × t, the horizontal distance from the launch point to the landing point.
- The highest point is H = h0 + vy² ÷ (2g), reached after vy ÷ g seconds.
- The impact speed is √(vx² + vy²) at landing, where the vertical speed is −√(vy² + 2g(h0 − h1)).
What the calculator does not cover
- Air resistance. A real ball or shell loses speed in the air, so its range is shorter than the ideal range shown here. Drag matters most for fast or light objects.
- Wind, spin and lift, which push the path sideways or change its shape.
- Changes of gravity with height over a long path, and the Earth's rotation, which matter mainly for long-range missiles and satellites.
Worked example: 20 m/s at 45° from the ground
The launch components are vx = v0 cos 45° = 14.14 m/s and vy = v0 sin 45° = 14.14 m/s. With h0 = h1 = 0, the time of flight is 2vy ÷ g = 28.28 ÷ 9.80665 = 2.884 s, and the range is v0² sin 2θ ÷ g = 400 ÷ 9.80665 = 40.79 m.
The highest point is vy² ÷ (2g) = 14.14² ÷ (2 × 9.80665) = 10.20 m, reached after vy ÷ g = 1.442 s. The projectile lands at 20 m/s, at 45° below the horizontal, because the launch and landing heights are equal.
| Quantity | Value |
|---|---|
| Range | 40.79 m |
| Time of flight | 2.884 s |
| Highest point | 10.2 m |
| Time to the highest point | 1.442 s |
| Speed and angle at impact | 20 m/s, 45° below the horizontal |
Frequently asked questions
Why does the range shrink in real life?
Air resistance slows the object in flight, so its range is shorter than the ideal range this calculator gives. The gap grows with speed, and it is large for light objects such as a feather or a light ball.
What launch angle gives the longest range?
Without air resistance and with equal launch and landing heights, 45° gives the longest range, because the range is v0² sin 2θ ÷ g. Two angles that add up to 90°, such as 30° and 60°, give the same range.
What if the object lands below the launch point?
Enter the landing height below the launch height. The flight is longer, because the object falls further, and the impact is steeper. The calculator uses the landing point you type, so a cliff or a pit gives the right range.
Why is the impact speed the same as the launch speed on level ground?
With no air resistance, energy is conserved: the speed squared depends only on the height drop. A projectile that lands at the height it started from has the same speed as at launch, but its direction is mirrored below the horizontal.
Can I enter the speed in km/h?
Yes. Type the speed with km/h, for example 72 km/h. The calculator converts it to metres per second, which is 20 m/s, before the working starts.
Does this work for a ball thrown by a person?
Roughly, if the air drag is small and the throw is short. A thrown ball loses speed to drag, so the real range is shorter. Use it to see how speed and angle change the path, not as an exact prediction.
Sources
- Wikipedia: Projectile motion (range and time of flight with the starting height)
- Wikipedia: Standard gravity (9.80665 m/s²)
Figures and text last checked on Tuesday, November 10, 2026.