Projectile motion calculator
This projectile motion calculator gives the range, maximum height and flight time from launch speed, angle and launch height, ignoring air resistance, on Earth or on the Moon, Mars and Jupiter.
Source: Wikipedia – Wurfparabel (Wurfweite, Wurfhöhe, optimaler Winkel). Updated: .
How it is calculated
Without air resistance a body follows a parabola: x = v₀·cos α·t, y = h₀ + v₀·sin α·t − g·t²/2. At 20 m/s and 45° on Earth that gives 40.8 m range, 10.2 m height and 2.88 s flight time.
How it is calculated
The speed is split into a horizontal part v₀·cos α and a vertical part v₀·sin α. The horizontal part stays constant, while gravity g slows the vertical part. The peak is at t = v₀·sin α / g, and the flight time is the positive root of the height y(t). Range, maximum height and impact speed follow from that.
Which angle gives the longest range?
From ground level, 45° is best. When you launch from a height h₀, the best angle is smaller: tan α = v₀ / √(v₀² + 2·g·h₀). The calculator shows that angle with its range. On the Moon the same throw goes about six times as far as on Earth.
- Angles above 45° give more height, angles below 45° put more speed into the distance.
- Two angles that add up to 90° (30° and 60°) give the same range from ground level.
Frequently asked questions
How do you calculate the range of a projectile?
From ground level, range = v₀² · sin(2α) / g. When launching from height h₀, range = v₀·cos α × flight time, where the flight time comes from y(t) = 0.
Which launch angle gives the longest range?
45° from ground level. With a launch height above zero the best angle is below 45°; the calculator shows it.
Does the calculator include air resistance?
No. It calculates the ideal throw in a vacuum. In air, range and height are smaller, noticeably so for fast or light objects.
Sources and legal basis
- Wikipedia – Wurfparabel (Wurfweite, Wurfhöhe, optimaler Winkel)
- Wikipedia – Projectile motion (range, height, time of flight)
- NASA Planetary Fact Sheet – surface gravitational acceleration of the celestial bodies
As of: · Editorial policy
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