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Range on level ground

The range is how far a throw goes when it lands at the same height as it started. Without air resistance it is largest at 45 degrees, because the sine of the doubled angle is then 1. An angle and its complement, for example 30 and 60 degrees, give the same range, but a different time in the air and a different maximum height. The formula does not apply if the ground slopes, or if the ball lands lower than the starting point.

R=v02sin⁡2αgR = \frac{v_0^2\sin 2\alpha}{g}range on level ground

Symbols

RRrangem
v0v_0initial speedm/s
α\alphaangle of projection°
ggacceleration of gravity, 9.81m/s²

Example

A ball is thrown at 20 m/s at 45° over level ground, without air resistance.

R=202sin⁡90∘9.81≈40.77R = \dfrac{20^2 \sin 90^\circ}{9.81} \approx 40.77 m.

In reality air resistance shortens the range, most for light balls and high speed.
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Part of Physics 2: Projectiles and motion in two dimensions.