When an object moves under the influence of a gravitational force (without air resistance), its path might be that of a parabola. This is the path of a ball thrown near the surface of a planet or other celestial object. Suppose two balls are simultaneously thrown upward at a $45^{\circ}$ angle on two different planets. If their initial velocities are both 30 mph, then their $x y$ -coordinates in feet can be expressed by the equation
$$y=x-\frac{g}{1922} x^{2}$$
where $g$ is the acceleration due to gravity. The value of $g$ will vary with the mass and size of the planet. (Source: Zeilik, M., S. Gregory, and E. Smith, Introductory Astronomy and Astrophysics, Saunders College Publishers.)
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(a) On Earth, $g=32.2$ and on Mars, $g=12.6 .$ Find the two equations, and use the same screen of a graphing calculator to graph the paths of the two balls thrown on Earth and Mars. Use the window $[0,180]$ by $[0,100].$
(b) Determine the difference in the horizontal distances traveled by the two balls.