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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. |
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