Shreya Kelly, Math Professor at Tohono Oodham Community College, Numerade Educator

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Calculus Early Transcendentals

If a projectile is fired with an initial velocity of $v_{0}$ meters per second at an angle $\alpha$ above the horizontal and air resistance is assumed to be negligible, then its position after $t$ seconds is given by the parametric equations
$$x=\left(v_{0} \cos \alpha\right) t \quad y=\left(v_{0} \sin \alpha\right) t-\frac{1}{2} g t^{2}$$
where $g$ is the acceleration due to gravity $\left(9.8 \mathrm{m} / \mathrm{s}^{2}\right).$
(a) If a gun is fired with $\alpha=30^{\circ}$ and $v_{0}=500 \mathrm{m} / \mathrm{s},$ when will the bullet hit the ground? How far from the gun will it hit the ground? What is the maximum height reached by the bullet?
(b) Use a graphing device to check your answers to part (a). Then graph the path of the projectile for several other values of the angle $\alpha$ to see where it hits the ground. Summarize your findings.
(c) Show that the path is parabolic by eliminating the parameter.

Chapter 10: Parametric Equations and Polar Coordinates

Section 1: Curves Defined by Parametric Equations