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Problem 54 Easy Difficulty

According to Newton's Law of Universal Gravitation, the gravitational force on an object of mass $ m $ that has been projected vertically upward from the earth's surface is
$ F = \frac {mgR^2}{(x + R)^2} $
where $ x = x(t) $ is the object's distance above the surface at time $ t, R $ is the earth's radius, and $ g $ is the acceleration due to gravity. Also, by Newton's Second Law, $ F = ma = m (dv/dt) $ and so
$ m \frac {dv}{dt} = - \frac {mgR^2}{(x + R)^2} $
(a) Suppose a rocket is fired vertically upward with an initial velocity $ v_0. $ Let $ h $ be the maximum height above

Answer

a) $V_{o}=\sqrt{\frac{2 g R h}{h+R}}$
b) $\sqrt{2 g R}$
c) $36580 f t / s, 6.93 \mathrm{mi} / \mathrm{s}$

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