cp's OEIS Frontend

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 expected length, averaged over theta uniformly chosen at random from the range [0, Pi/2], is c * v^2/g, where c is this constant.

The length of the trajectory as a function of theta is L(theta) = (v^2/g)*(sin(theta) + cos(theta)^2*log((1+sin(theta))/(1-sin(theta)))/2). L(theta) goes from 0 to 1 between theta = 0 and Pi/2. It has a maximum at theta = 0.985514... (A345737), and a unique value at 0 <= theta < 0.599677... (A345739). The average length (c * v^2/g) occurs at theta = 0.5152731296... (29.522975... degrees).

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.