March 2001
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On the first hole, suppose the ball's initial speed is (v), at an initial angle (b) up from the horizontal. Its horizontal velocity is then (v cos b). Its initial vertical velocity is (v sin b). Letting (g) be the acceleration due to gravity, we see that after time (t = 2 v sin b / g) the vertical velocity will have changed from (v sin b) to (- v sin b), and the ball will have returned to its initial vertical position. Let (D=150 yards) denote the horizontal distance of this hole, and observe that in time (t) the ball has travelled a horizontal distance of (t v cos b), leading to the equation:
D = (v^2 /g) 2 sin b cos b,
or
D = (v^2 /g) sin(2b).
To maximize the horizontal distance for a given initial speed (v), we must maximize (sin(2b)), which we do by setting (b=45 degrees), whence (sin(2b)=1), and
v = sqrt ( D g ).
On the third hole, denote the initial speed as V, the initial angle above the horizon as B, and the total time as T.
The horizontal distance will be
D / sqrt(2) = T V cos B.
The average vertical velocity will be
V sin B - (T/2) g,
so that the total vertical ascent will be
- D / sqrt(2) = T ( V sin B - (T/2) g ).
Solving for (V):
V^2 = D g / ( 2 sqrt(2) * (cos B) * (sin B + cos B) ).
Use the identity:
(cos B)*(sin B + cos B) = (1/2) + (1/sqrt(2))*sin(2B+45)
to deduce that (V) will be minimized when (B=22.5 degrees), with a ratio
V/v = sqrt ( 1 - 1/sqrt(2) ) = 0.5412.
On the third hole, the golfer hits the ball at 54.12% of the speed that was required on the first hole.
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