cp's OEIS Frontend

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.

%I A345739 #11 Apr 13 2025 01:46:49
%S A345739 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,
%T A345739 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,
%U A345739 3,8,4,8,2,4,4,3,3,6,9,3,5,6,6,3,6,8,1
%N 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.
%C A345739 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)).
%C A345739 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).
%C A345739 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).
%C A345739 Equals 34.3590116998... degrees.
%H A345739 Haiduke Sarafian, <a href="https://doi.org/10.1119/1.880184">On projectile motion</a>, The Physics Teacher, Vol. 37, No. 2 (1999), pp. 86-88.
%e A345739 0.59967788189342845872847593688137561917190103459359...
%t A345739 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]]
%Y A345739 Cf. A345737, A345738.
%K A345739 nonn,cons
%O A345739 0,1
%A A345739 _Amiram Eldar_, Jun 25 2021