A ski jumper starts with negligible speed from height H and…

A ski jumper starts with negligible speed from height $H$ and leaves the jump horizontally at height $h$ above the ground. What height $h$ of the take-off gives the longest jump (measured horizontally from the foot of the jump), and how long is it? Neglect friction and air resistance.

Explanation of Method

Take-off height for the longest ski jump.

Step 1: Take-off speed and flight time

\[ v = \sqrt{2g\,(H - h)}, \]

\[ t = \sqrt{\frac{2h}{g}}. \]

Step 2: Jump length

\[ s = vt = 2\sqrt{h\,(H - h)}. \]

The product $h(H - h)$ is largest when the two factors are equal: $h = \frac{H}{2}, \qquad s_{\max} = H$.

$H$ $h = \tfrac{H}{2}$ $s_{\max} = H$ $s = 2\sqrt{h(H - h)}$

Drawn to scale for the best take-off height $h = H/2$: the jumper leaves horizontally and lands exactly $H$ from the foot of the jump.

Result

\[\boxed{\,h = \frac{H}{2}, \qquad s_{\max} = H\,}\]

Remark

Check. The original results are correct.