Chapter 25, Problem 22P

Suppose that a projectile is launched upward from the earth's surface. Assume that the only force acting on the object is the downward force of gravity. Under these conditions, a force balance can be used to derive,

$\frac{dv}{dt}=-g\left(0\right)\frac{R^2}{\left(R+x^2\right)}$

where $v=$ upward velocity (m/s), $t=$ time (s), $x=$ altitude (m) measured upwards from the earth's surface, $g\left(0\right)=$ the gravitational acceleration at the earth's surface $\left(\cong 9.81{\text{ m/s}}^2\right)$, and $R=$ the earth's radius $\left(\cong 6.37\times {10}^6\text{ m}\right)$. Recognizing that $dx/dt=v$, use Euler's method to determine the maximum height that would be obtained if $v\left(t=0\right)=1500\text{ m/s}$.