+0

# help

0
65
1

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 $$c.$$

Nov 11, 2019