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A345738
Decimal expansion of (2*G+1)/Pi, where G is Catalan's constant (A006752).
2
9, 0, 1, 4, 3, 1, 6, 9, 4, 2, 4, 5, 4, 2, 8, 2, 3, 1, 8, 1, 4, 5, 3, 6, 4, 3, 9, 6, 8, 1, 8, 1, 8, 5, 6, 1, 7, 9, 7, 0, 5, 1, 5, 9, 9, 4, 5, 2, 5, 8, 7, 4, 3, 8, 0, 1, 7, 3, 3, 7, 8, 2, 6, 3, 4, 1, 2, 8, 8, 8, 6, 9, 0, 2, 9, 3, 3, 0, 7, 9, 3, 6, 3, 3, 4, 8, 1
OFFSET
0,1
COMMENTS
A projectile is launched with an initial speed v at angle theta above the horizon. Assuming that the gravitational acceleration g is uniform and neglecting the air resistance, the trajectory is a part of a parabola whose expected length, averaged over theta uniformly chosen at random from the range [0, Pi/2], is c * v^2/g, where c is this constant.
The length of the trajectory as a function of theta is L(theta) = (v^2/g)*(sin(theta) + cos(theta)^2*log((1+sin(theta))/(1-sin(theta))/2. L(theta) goes from 0 to 1 between theta = 0 and Pi/2. It has a maximum at theta = 0.985514... (A345737), and a unique value at 0 <= theta < 0.599677... (A345739). The average length (c * v^2/g) occurs at theta = 0.5152731296... (29.522975... degrees).
LINKS
Péter Kórus, Notes on Projectile Motion, The American Mathematical Monthly, Vol. 126, No. 4 (2019), pp. 358-360.
FORMULA
Equals (2 * A006752 + 1)/A000796.
Equals 2 * A143233 + 1.
EXAMPLE
0.90143169424542823181453643968181856179705159945258...
MATHEMATICA
RealDigits[(2*Catalan + 1)/Pi, 10, 100][[1]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Jun 25 2021
STATUS
approved