Chapter 4.3, Problem 15P

Suppose that a projectile is fired straight upward with initial velocity $v_0$ from the surface of the earth. If air resistance is not a factor, then its height $x\left(t\right)$ at time t satisfies the initial value problem $\frac{d^{2}x}{d{t}^2}=-\frac{g{R}^2}{{\left(x+R\right)}^2};x\left(0\right)=0,x\text{'}\left(0\right)=v_0.$

Use the values $g=32.15{\text{ft/s}}^{\text{2}}\approx 0.006089{\text{mi/s}}^{\text{2}}$ for the gravitational acceleration of the earth at its surface and $R=3960\text{mi}$ as the radius of the earth. If $v_0=1\text{mi/s,}$ find the maximum height attained by the projectile and its time of ascent to this height.