# Parabolas of Safety

Suppose a projectile is thrown from the center of the coordinate system $O(0, 0)$ with initial velocity $v_0$ at angle $\alpha$ with the x -axis. The motion of the projectile in vacuum is demonstrated by the applet below. When the angle $\alpha$ changes randomly from 0 to $\pi$ we obtain a family of parabolas with the same initial velocity. The envelope of the family is a parabola with vertex $V=\left(0, \frac{v_0\,^2}{2g}\right )$, focus at the origin $O(0,0)$, and $x$ - intercepts $x=\pm \frac{v_0\,^2}{g}= \pm x_{max}$. [list] [*]To simulate the motion, click the "Animate" button or dragg slider n. [*]Click the "Show Foci" checkbox to see the focus of the parabolas as they are constructed. [*]Click the "Locus of the Foci" checkbox to see the locus of the foci of the parabolas. [/list]