Parabolas of Safety

Suppose a projectile is thrown from the center of the coordinate system [math]O(0, 0)[/math] with initial velocity [math]v_0[/math] at angle [math]\alpha[/math] with the x -axis. The motion of the projectile in vacuum is demonstrated by the applet below. When the angle [math]\alpha[/math] changes randomly from 0 to [math]\pi [/math] we obtain a family of parabolas with the same initial velocity. The envelope of the family is a parabola with vertex [math]V=\left(0, \frac{v_0\,^2}{2g}\right ) [/math], focus at the origin [math]O(0,0) [/math], and [math]x[/math] - intercepts [math]x=\pm \frac{v_0\,^2}{g}= \pm x_{max}[/math]. [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]