A satellite of mass m, originally on the surface of the Earth, is placed into Ea

Question

A satellite of mass m, originally on the surface of the Earth, is placed into Earth orbit at an altitude h. (Use any variable or symbol stated above along with the following as necessary: RE for the radius of the Earth, ME for the mass of the Earth, G for the gravitational constant, and ?.) (a) Assuming a circular orbit, how long does the satellite take to complete one orbit? T = Incorrect: Your answer is incorrect. (b) What is the satellite's speed? v = Correct: Your answer is correct. (c) What is the minimum energy input necessary to place this satellite in orbit? Ignore air resistance but include the effect of the planet's daily rotation. ?Emin

Explanation / Answer

The initial energy (on the ground) is potential; Ei = - Gmme/r The final energy is potential plus kinetic; Ef = -Gmme/(r+h) + (1/2)mv^2 The energy that must be supplied (work done) is equal to the change in energy; W = (Ef - Ei) W = -Gmme/(r+h) + (1/2)mv^2 + Gmme/r The Kinetic energy term can be expressed in terms of the radial distances by requiring the centripetal force equal the gravitational force; mv^2/(r+h) = Gmme/(r+h)^2 mv^2 = Gmme/(r+h) So now the energy supplied can be written as; W = -Gmme/(r+h) + Gmme/2(r+h) + Gmme/r Add these terms together by finding a common denominator and the answer will fall out

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