How this calculator works
- Step 1Splitv cos θ, v sin θHorizontal and vertical speeds.
- Step 2Timevertical motionGravity decides how long it stays up.
- Step 3Rangevₓ × tHorizontal speed never changes.
A projectile’s motion splits into two parts. Horizontally nothing pushes or slows it, so its speed stays constant. Vertically, gravity slows it on the way up and speeds it on the way down.
The flight time comes from the vertical motion alone. Multiply it by the horizontal speed to get the range.
Worked example: 20 m/s at 45°
- t
(v_y + √(v_y² + 2gh)) ÷ g- R
vₓ × t
- Horizontal speed20 × cos 45°14.14 m/s
- Vertical speed20 × sin 45°14.14 m/s
- Flight time2 × 14.14 ÷ 9.82.89 s
- Maximum height14.14² ÷ (2 × 9.8)10.20 m
- Range14.14 × 2.8940.82 m
Read the result
From ground level, 45° gives the longest range, and angles that add to 90° give the same range.
- On the Moon246.91 mWeaker gravity keeps it up six times longer.
- Level from a 20 m cliff30.30 m15 m/s, falling for 2.02 s.
Assumptions and limits
- No airDrag and wind are ignored.
- Flat groundIt lands at height 0.
- Constant gravityg does not change with height.
- No spinSpin and lift are not modelled.
Related motion tools
A projectile is constant acceleration in two directions at once.
For straight-line motion with constant acceleration, use the kinematics calculator. The kinetic energy calculator gives the energy at launch, and the speed calculator handles average speed.
Common mistakes
Mistake: Entering km/h as m/s.
Instead: Divide km/h by 3.6.
Mistake: Using 45° from a height.
Instead: From above the ground a lower angle reaches further.
Mistake: Expecting these ranges for real balls.
Instead: Air resistance cuts them, especially at speed.
Checking the method
A separate script simulates every flight in tiny time steps, without the flight formulas, and matches each golden value.
Read how we test our calculators, or browse more maths calculators.