|
|
A345737
|
|
Decimal expansion of the initial angle in radians above the horizon that maximizes the length of a projectile's trajectory.
|
|
2
|
|
|
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
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,1
|
|
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.
|
|
REFERENCES
|
Thomas Szirtes, Applied Dimensional Analysis and Modeling, Butterworth-Heinemann, 2007, p. 578.
|
|
LINKS
|
|
|
FORMULA
|
Equals arccsc(u) where u is the root of coth(x) = x (A085984).
|
|
EXAMPLE
|
0.98551473786231546211492853725730463877247220596742...
|
|
MATHEMATICA
|
RealDigits[ArcCsc[x /. FindRoot[x == Coth[x], {x, 1}, WorkingPrecision -> 120]], 10, 100][[1]]
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|