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 −

mtR2 sx 1 Rd2 where x − xstd is the object’s distance above the surface at time t, R is the earth’s radius, and t is the acceleration due to gravity. Also, by Newton’s Second Law, F − ma − msdvydtd and so m

dv dt

− 2 mtR2 sx 1 Rd2

(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 v0 −Î 2tRh R 1 h

[Hint: By the Chain Rule, msdvydtd − mv sdvydxd.]

(b) Calculate ve − limh l ` v0. This limit is called the escape velocity for the earth.

(c) Use R − 3960 mi and t − 32 ftys2 to calculate ve in feet per second and in miles per second.