A projectile is fired with an initial velocity of \(v\) at an angle of \(\theta\) from the ground. Then its trajectory can modeled by the parametric equations \(\begin{align*} x &= vt \cos \theta, \\ y &= vt \sin \theta - \frac{1}{2} gt^2, \end{align*}\)
where \(t\) denotes time and \(g\) denotes acceleration due to gravity, forming a parabolic arch.
Suppose \(v\) is held constant, but \(\theta\) is allowed to vary, over \(0^\circ \le \theta \le 180^\circ.\) The highest point of each parabolic arch is plotted. As \(\theta\) varies, the highest points of the arches trace a closed curve. The area of this closed curve can be expressed in the form \(c \cdot \frac{v^4}{g^2}.\)
Find \(c.\)
