A345738 - OEIS

%I #8 Jun 26 2021 02:16:27

%S 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,

%T 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,

%U 8,8,6,9,0,2,9,3,3,0,7,9,3,6,3,3,4,8,1

%N Decimal expansion of (2*G+1)/Pi, where G is Catalan's constant (A006752).

%C 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.

%C 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).

%H Péter Kórus, <a href="https://doi.org/10.1080/00029890.2019.1565858">Notes on Projectile Motion</a>, The American Mathematical Monthly, Vol. 126, No. 4 (2019), pp. 358-360.

%F Equals (2 * A006752 + 1)/A000796.

%F Equals 2 * A143233 + 1.

%e 0.90143169424542823181453643968181856179705159945258...

%t RealDigits[(2*Catalan + 1)/Pi, 10, 100][[1]]

%Y Cf. A000796, A006752, A143233, A345737, A345739.

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, Jun 25 2021