This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A345737 #22 Jun 22 2025 23:11:46
%S A345737 9,8,5,5,1,4,7,3,7,8,6,2,3,1,5,4,6,2,1,1,4,9,2,8,5,3,7,2,5,7,3,0,4,6,
%T A345737 3,8,7,7,2,4,7,2,2,0,5,9,6,7,4,2,9,6,4,8,1,2,7,8,4,5,1,1,4,0,3,2,8,2,
%U A345737 9,5,2,7,0,5,2,0,8,0,5,3,5,7,2,5,7,1,5
%N A345737 Decimal expansion of the initial angle in radians above the horizon that maximizes the length of a projectile's trajectory.
%C A345737 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 maximized when the angle is the root of the equation csc(theta) = coth(csc(theta)). The maximal length is then u * v^2/g, where u = 1.1996... is the root of coth(x) = x (A085984).
%C A345737 The angle in degrees is 56.4658351274...
%C A345737 The initial angle that maximizes the horizontal distance is the well-known result theta = Pi/4 = 45 degrees. The corresponding length of trajectory in this case is u * v^2/g, where u = (sqrt(2) + arcsinh(1))/2 = 1.1477... (A103711), which is 95.67...% of the maximum value.
%D A345737 Thomas Szirtes, Applied Dimensional Analysis and Modeling, Butterworth-Heinemann, 2007, p. 578.
%H A345737 Joshua Cooper and Anton Swifton, <a href="https://www.jstor.org/stable/10.4169/amer.math.monthly.124.10.955">Throwing a ball as far as possible, revisited</a>, The American Mathematical Monthly, Vol. 124, No. 10 (2017), pp. 955-959; <a href="https://arxiv.org/abs/1611.02376">arXiv preprint</a>, arXiv:1611.02376 [math.HO], 2016.
%H A345737 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.
%H A345737 Ju Yan-Qing, <a href="https://en.cnki.com.cn/Article_en/CJFDTotal-GLKX200503018.htm">Projectile motion path length and initial projectile angle</a>, Journal of Science of Teachers' College and University, Vol. 3 (2005), pp. 49-51.
%F A345737 Equals arccsc(u) where u is the root of coth(x) = x (A085984).
%F A345737 Equals arctan(A240358) = arctan(1/A033259). - _Robert FERREOL_, Jun 16 2025
%F A345737 Positive root of tan(x) = sinh(csc(x)). - _Robert FERREOL_, Jun 17 2025
%e A345737 0.98551473786231546211492853725730463877247220596742...
%p A345737 Digits:=100:fsolve(tan(x)=sinh(csc(x)),x=0..1); (# _Robert FERREOL_, Jun 17 2025)
%t A345737 RealDigits[ArcCsc[x /. FindRoot[x == Coth[x], {x, 1}, WorkingPrecision -> 120]], 10, 100][[1]]
%o A345737 (PARI) solve(x=0,1,my(s=sin(x)); s*atanh(s)-1) \\ _Charles R Greathouse IV_, Sep 18 2024
%o A345737 (PARI) asin(solve(u=.5, 1, tanh(1/u)-u)) \\ _Charles R Greathouse IV_, Sep 18 2024
%Y A345737 Cf. A033259, A085984, A103711, A240358, A345738, A345739.
%K A345737 nonn,cons
%O A345737 0,1
%A A345737 _Amiram Eldar_, Jun 25 2021