How this calculator works
Horizontal speed stays constant. The vertical equation determines landing time, and range is horizontal speed times that time. Starting above the landing plane can increase range and makes the optimum launch angle lower than 45 degrees. A target range with a specified speed may allow two launch angles: a flatter path and a steeper path. Both angles are reported when distinct and reachable.
Formula
T = (vy + sqrt(vy^2+2gh0))/g; R = vx T. For h0=0, R = v^2 sin(2alpha)/g.
Worked example
At g = 10 m/s2, a 10 m/s launch with components 8 m/s horizontal and 6 m/s upward stays aloft 1.2 s and travels 9.6 m. Reversing the calculation recovers the 10 m/s launch speed.
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Frequently asked questions
Is 45 degrees always best?
It maximizes range only when launch and landing heights match, gravity is uniform and drag is absent.
Why can there be two angles?
A shallow fast flight and a high slow flight can land at the same horizontal distance.
Can I use this for wind or spin?
No. Those effects change the flight path and require additional force models.