Serving the ball

Worked example 1: the serving window. A seated server hits the ball 1.3 m above the floor, 0.5 m behind the end line, at 15° above the horizontal (illustrative). The net is 5.5 m away and its top is at 1.15 m, so the ball's centre must be at least $1.15+0.105=1.255\mathrm{m}1.15\; +\; 0.105\; =\; 1.255\backslash ,\backslash mathrm\{m\}$ high there (the ball's radius is about 0.105 m). The far end line is 10.5 m away. For a projectile,

1. Slowest serve that clears the net: $v^2=\frac{9.81\times {5.5}^2}{2{\cos }^2{15}^{\circ }(1.3+5.5\tan {15}^{\circ }-1.255)}v^\{2\}\; =\; \backslash frac\{9.81\; \backslash times\; 5.5^\{2\}\}\{2\backslash cos^\{2\}15^\{\backslash circ\}\backslash ,(1.3\; +\; 5.5\backslash tan\; 15^\{\backslash circ\}\; -\; 1.255)\}$, so $v=10.2\mathrm{m}{\mathrm{s}}^{-1}v\; =\; 10.2\backslash ,\backslash mathrm\{m\backslash ,s^\{-1\}\}$.
2. Fastest serve that still lands in: $v^2=\frac{9.81\times {10.5}^2}{2{\cos }^2{15}^{\circ }(1.3+10.5\tan {15}^{\circ }-0.105)}v^\{2\}\; =\; \backslash frac\{9.81\; \backslash times\; 10.5^\{2\}\}\{2\backslash cos^\{2\}15^\{\backslash circ\}\backslash ,(1.3\; +\; 10.5\backslash tan\; 15^\{\backslash circ\}\; -\; 0.105)\}$, so $v=12.0\mathrm{m}{\mathrm{s}}^{-1}v\; =\; 12.0\backslash ,\backslash mathrm\{m\backslash ,s^\{-1\}\}$.

Any speed between 10.2 and 12.0 m s⁻¹ goes in: a window of only 1.8 m s⁻¹. At 10° the window is 12.3–13.5 m s⁻¹, only 1.2 m s⁻¹ wide, and at 5° there is no window at all. Because the server sits so low, even a low net demands an upward serve, and a consistent technique matters more than raw power.

Worked example 2: rating a server. In a class tournament, Mutoni serves 20 times: 3 aces, 14 serves in play and 3 errors. Gasana serves 20 times: 6 aces, 8 in play and 6 errors (illustrative).

1. Success rate (serves not lost): Mutoni $\frac{17}{20}=85\mathrm{\%}\backslash frac\{17\}\{20\}\; =\; 85\backslash \%$; Gasana $\frac{14}{20}=70\mathrm{\%}\backslash frac\{14\}\{20\}\; =\; 70\backslash \%$.
2. Efficiency $=\frac{\text{aces}-\text{errors}}{\text{serves}}=\; \backslash frac\{\backslash text\{aces\}\; -\; \backslash text\{errors\}\}\{\backslash text\{serves\}\}$: Mutoni $\frac{3-3}{20}=0\backslash frac\{3\; -\; 3\}\{20\}\; =\; 0$; Gasana $\frac{6-6}{20}=0\backslash frac\{6\; -\; 6\}\{20\}\; =\; 0$.

Both servers win as many points directly as they give away. Gasana's aggressive serve wins more aces but gives away more points; Mutoni's keeps the ball in and lets the team play rallies. A team usually wants reliable servers, with a strong serve kept for moments when an ace matters most.