An object is projected vertically upward from the Earth's surface with an initial velocity of $v_0($mph$)$ of magnitude less than the escape velocity. Escape velocity is defined as the velocity that a projectile should have to break free of the Earth forever. If only the Earth's influence $($gravity$)$ is taken into consideration, then the maximum height, in miles, reached by the projectile is:

$H(v_0)=\frac{v_0^{2}R}{2gR-v_0^2},v_{0}0$

where $R$ is the radius of the Earth $($in miles$)$ and $g$ is acceleration of gravity $($miles $\frac{?}{h{r}^2}).$

a$.$ Determine the vertical asymptote of $H(v_0)$ in terms $R$ and$/$or $g$

b$.$ Interpret your results from part $($a$)$ based on the context of this scenario

c$.$ Find the escape velocity in miles per hour. Hint: Earth's radius is about $4,000$ miles, $5,280$ feet in $1$ mile, and $g=32f\frac{t}{se{c}^2}$

d$.$ Sketch the graph of $H(v_0)$ and clearly label the vertical asymptote in terms of $R$ and$/$or $g$