A345739 Decimal expansion of the initial acute angle in radians above the horizon of a projectile's velocity such that length of its trajectory is equal to the length of the vertical trajectory with the same initial speed.

5, 9, 9, 6, 7, 7, 8, 8, 1, 8, 9, 3, 4, 2, 8, 4, 5, 8, 7, 2, 8, 4, 7, 5, 9, 3, 6, 8, 8, 1, 3, 7, 5, 6, 1, 9, 1, 7, 1, 9, 0, 1, 0, 3, 4, 5, 9, 3, 5, 9, 6, 7, 9, 6, 0, 6, 8, 0, 0, 0, 3, 5, 3, 7, 6, 5, 5, 6, 3, 8, 4, 8, 2, 4, 4, 3, 3, 6, 9, 3, 5, 6, 6, 3, 6, 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 length is L(theta) = (v^2/g) * f(theta), where f(x) = sin(x) + cos(x)^2*log((1+sin(x))/(1-sin(x)))/2 = sin(x) + cos(x)^2*log((1+sin(x))/cos(x)).

This constant is the smaller of the two real roots of f(x) = 1 (the larger root is x = Pi/2, corresponding to a vertical trajectory).

At this angle the trajectory's length is equal to v^2/g, which is also the length of the trajectory at vertical initial velocity (i.e., twice the maximum height). For angles theta below this constant L(theta) has a unique value (i.e., not shared with any other angle).

Equals 34.3590116998... degrees.

Examples

			0.59967788189342845872847593688137561917190103459359...
		

Links

• Haiduke Sarafian, On projectile motion, The Physics Teacher, Vol. 37, No. 2 (1999), pp. 86-88.

Crossrefs

Cf. A345737, A345738.

Programs

Mathematica

RealDigits[x /. FindRoot[Sin[x] + (1/2)*Cos[x]^2*Log[(1 + Sin[x])/(1 - Sin[x])] - 1, {x, 1/2}, WorkingPrecision -> 120], 10, 100][[1]]