A345737 Decimal expansion of the initial angle in radians above the horizon that maximizes the length of a projectile's trajectory.

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, 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, 9, 5, 2, 7, 0, 5, 2, 0, 8, 0, 5, 3, 5, 7, 2, 5, 7, 1, 5

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 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).

The angle in degrees is 56.4658351274...

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.

Examples

			0.98551473786231546211492853725730463877247220596742...
		

References

• Thomas Szirtes, Applied Dimensional Analysis and Modeling, Butterworth-Heinemann, 2007, p. 578.

Links

• Joshua Cooper and Anton Swifton, Throwing a ball as far as possible, revisited, The American Mathematical Monthly, Vol. 124, No. 10 (2017), pp. 955-959; arXiv preprint, arXiv:1611.02376 [math.HO], 2016.
• Haiduke Sarafian, On projectile motion, The Physics Teacher, Vol. 37, No. 2 (1999), pp. 86-88.
• Ju Yan-Qing, Projectile motion path length and initial projectile angle, Journal of Science of Teachers' College and University, Vol. 3 (2005), pp. 49-51.

Crossrefs

Cf. A033259, A085984, A103711, A240358, A345738, A345739.

Programs

Maple

Digits:=100:fsolve(tan(x)=sinh(csc(x)),x=0..1); (# Robert FERREOL, Jun 17 2025)

Mathematica

RealDigits[ArcCsc[x /. FindRoot[x == Coth[x], {x, 1}, WorkingPrecision -> 120]], 10, 100][[1]]

PARI

solve(x=0,1,my(s=sin(x)); s*atanh(s)-1) \\ Charles R Greathouse IV, Sep 18 2024

PARI

asin(solve(u=.5, 1, tanh(1/u)-u)) \\ Charles R Greathouse IV, Sep 18 2024

Formula

Equals arccsc(u) where u is the root of coth(x) = x (A085984).

Equals arctan(A240358) = arctan(1/A033259). - Robert FERREOL, Jun 16 2025

Positive root of tan(x) = sinh(csc(x)). - Robert FERREOL, Jun 17 2025