Mark:
1934 DISMISSAL
Edmund Landau
It is almost unbelievable that a definition of pi was used, at least as an excuse, for a racial attack on the eminent mathematician Edmund Landau in 1934. Landau had defined pi in this textbook published in Göttingen in that year by the, now fairly usual, method of saying that pi/2 is the value of x between 1 and 2 for which cos x vanishes. This unleashed an academic dispute which was to end in Landau's dismissal from his chair at Göttingen.
Mark:
GOLF
First ball: angle = 45 degrees, speed = 120 ft/sec, elapsed time = 5.3033 seconds
Second ball: angle = 22.5 degrees, speed = 64.9435 ft/sec, elapsed time = 5.3033 seconds
64.9435/120 = a bit more than 54%
Notice that the balls arrive in the cups at the same time.
Well done people, we won the cup. (Nice putt Mark)
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.
The physicist Willebrord Snell (1580-1626) found polygons which better approximated the perimeter of circles than do inscribed and circumscribed polygons. Better perimeter approximations lead to more quickly converging pi approximations. What scientific discovery is Snellius best known for
the laws of reflection and refraction
general relativity
exploding pop-tarts
the photoelectric effect
the uncertainty principle in quantum mechanics
How many zeros are at the end of 100000!
