Chapter 10.1, Problem 40ES

We will show later in the text that if a projectile is fired from ground level with an initial speed of ${\upsilon }_0$ meters per second at an angle $\alpha$ with the horizontal, and if air resistance is neglected, then its position after t seconds, relative to the coordinate system in the accompanying figure is $x=\left({\upsilon }_0\cos \alpha \right)t,y=\left({\upsilon }_0\sin \alpha \right)t-\frac{1}{2}g{t}^2$ where $g\approx 9.8m/{\text{s}}^2.$

(a) By eliminating the parameter, show that the trajectory lies on the graph of a quadratic polynomial.

(b) Use a graphing utility to sketch the trajectory if $\alpha =30°$ and ${\upsilon }_o=1000m/\text{s}.$

(c) Using the trajectory in part (b), how high does the shell rise?

(d) Using the trajectory in part (b), how far does the shell travel horizontally?