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
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 lo gravity. Also, by Newton’s Second Law, F = ma = m(dv/dt) and so
(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
[Hint: By the Chain Rule, m(dv/dt) = mv(dv/dx).]
(b) Calculate ve = limh→∞v0. This limit is called the escape velocity for the 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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Essential Calculus: Early Transcendentals
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning