How to use this calculator
- Enter or choose launch speed.
- Enter or choose speed unit.
- Enter or choose launch angle above horizontal.
- Enter or choose launch height above landing level.
- Enter or choose height unit.
- The horizontal range updates as you type. Use Copy result or Share to keep it.
Method
t = (v sinθ + √((v sinθ)² + 2 g h)) ÷ g; R = v cosθ × t; on level ground R = v² sin 2θ ÷ g
- R
- horizontal range in m
- v
- launch speed in m/s
- θ
- launch angle above horizontal
- h
- launch height above the landing level in m
- g
- gravitational acceleration
- t
- time of flight in s
Worked example
- Launched horizontally (θ = 0) from 20 m with g = 10 m/s²: t = √(2h ÷ g) = √(40 ÷ 10) = 2 s.
- R = 10 m/s × 2 s = 20 m.
A ball thrown horizontally at 10 m/s from 20 m up lands 20 m away after 2 seconds (with g rounded to 10 m/s²).
What the result means
On level ground the range is greatest at 45°, and complementary angles (30° and 60°) give the same range. Launching from a height shifts the best angle below 45°. The table shows how range changes with angle for your speed.
Assumptions and limits
- No air resistance, spin or wind.
- Flat landing surface at the stated level and constant g.
- Real balls, arrows and shells fall well short of the vacuum range at high speed because drag grows with the square of speed.
Questions
Why is 45° the best angle?
Range on level ground is v² sin 2θ ÷ g, and sin 2θ peaks at 1 when θ = 45°.
Does mass affect the range?
Not without air resistance. With drag, heavier and denser objects of the same shape keep their speed better and travel farther.