According to Newton€™s Law of Universal Gravitation, the gravitational force on an object of mass m that has been projected vertically upward from Earth€™s surface is F = mgR2/(x + R)2 where x = x(t) is the object€™s distance above the surface at time t, R is Earth€™s radius, and is the acceleration due to gravity.
Also, by Newton€™s Second Law, F = ma = m (dv/dt) and so m dv/dt = €“ mgR2 / (x + R) 2.
(a) Suppose a rocket is fired vertically upward with an initial velocity v0. Let h be the maximum height above the surface reached by the object. Show that




(b) Calculate ve = lim h→∞ v0. This limit is called the escape velocity for Earth.
(c) Use R = 3960 mi and g = 32 ft/s2 to calculate ve in feet per second and in miles per second.

Transcribed Image Text:

2gRh VR + h