A projectile is launched from ground level at an angle of $\theta$ with the horizontal. The initial velocity is $v _ { 0 }$ feet per second and the path of the projectile is modeled by the parametric equations $x = \left( v _ { 0 } \cos \theta \right) t$ and $y = \left( v _ { 0 } \sin \theta \right) t - 16 t ^ { 2 } .$
Use a graphing utility to graph the paths of a projectile launched from ground level with the values given for $\theta$ and $v _ { 0 }$. Use the graph to approximate the maximum height and range of the projectile to the nearest foot.
$$\theta = 50 ^ { \circ } , \quad v _ { 0 } = 104 \text { feet per second }$$
A) maximum height $\approx 12$ feet range $\approx 171$ feet
B) maximum height $\approx 171$ feet range $\approx 12$ feet
C) maximum height $\approx 99$ feet range $\approx 333$ feet
D) maximum height $\approx 333$ feet $\quad$ range $\approx 99$ feet
E) maximum height $\approx 38$ feet range $\approx 8$ feet

Question

Quizplus · Accepted Answer

The maximum height and range of a projectile can be calculated using the given parametric equations and the initial conditions. For a projectile launched at an angle $\theta$ with an initial velocity $v_0$, the maximum height $H$ can be found using the formula $H = \frac{v_0^2 \sin^2\theta}{2g}$, and the range $R$ can be calculated using $R = \frac{v_0^2 \sin 2\theta}{g}$, where $g$ is the acceleration due to gravity (approximately $32 \text{ ft/s}^2$ in this context). Given $\theta = 50^\circ$ and $v_0 = 104 \text{ ft/s}$, the calculations approximate to a maximum height of about $99$ feet and a range of about $333$ feet.

A Projectile Is Launched from Ground Level at an Angle θ\thetaθ

A Projectile Is Launched from Ground Level at an Angle $\theta$