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 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.
LINKS
Haiduke Sarafian, On projectile motion, The Physics Teacher, Vol. 37, No. 2 (1999), pp. 86-88.
EXAMPLE
0.59967788189342845872847593688137561917190103459359...
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]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Jun 25 2021
STATUS
approved