A345738 Decimal expansion of (2*G+1)/Pi, where G is Catalan's constant (A006752).

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

Author

Amiram Eldar, Jun 25 2021

Keywords

• nonn
• cons

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).

Examples

			0.90143169424542823181453643968181856179705159945258...
		

Links

• Péter Kórus, Notes on Projectile Motion, The American Mathematical Monthly, Vol. 126, No. 4 (2019), pp. 358-360.

Crossrefs

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

Programs

Mathematica

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

Formula

Equals (2 * A006752 + 1)/A000796.

Equals 2 * A143233 + 1.