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.
EXAMPLE
0.90143169424542823181453643968181856179705159945258...
MATHEMATICA
RealDigits[(2*Catalan + 1)/Pi, 10, 100][[1]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Jun 25 2021
STATUS
approved