Chapter 4, Problem 18P

A missile leaves the ground with an initial velocity $v_0$ forming an angle ${\varphi }_0$ with the vertical as shown in Fig. P4.18. The maximum desired altitude is $\alpha R$ where R is the radius of the earth. The laws of mechanics can be used to show that

$\sin {\varphi }_0=\left(1+\alpha \right)\sqrt{1-\frac{\alpha }{1+\alpha }{\left(\frac{v_e}{v_0}\right)}^2}$

where $v_e=$ the escape velocity of the missile. It is desired to fire the missile and reach the design maximum altitude within an accuracy of 62%. Determine the range of values for $\pm 2\%$. Determine the range of values for ${\varphi }_0\text{ if }{v}_e/v_0=2\text{ and }\alpha =0.25$

FIGURE P4.18