Fundamentals of Physics Extended (10th Edition)

Published by Wiley
ISBN 10: 1-11823-072-8
ISBN 13: 978-1-11823-072-5

Chapter 13 - Gravitation - Problems - Page 383: 62d

Answer

Satellite B has a greater total energy than satellite A by $~~1.14\times 10^8~J$

Work Step by Step

Let $R_E$ be the Earth's radius. The total energy of an orbiting satellite at a height $h$ above the Earth's surface is $E = -\frac{GMm}{2(R_E+h)}$ We can see that as the value of $h$ increases, the total energy becomes less negative. (That is, the magnitude decreases as the value moves closer to zero.) Since satellite B is at a higher altitude than satellite A, satellite B has a greater total energy than satellite A. We can find the difference in the total energy of each satellite: $E_B-E_A = -\frac{GMm}{2(R_E+h_B)}- [-\frac{GMm}{2(R_E+h_A)}]$ $E_B-E_A = \frac{GMm}{2(R_E+h_A)} -\frac{GMm}{2(R_E+h_B)}$ $E_B-E_A = \frac{(6.67\times 10^{-11}~N~m^2/kg^2)(5.98\times 10^{24}~kg)(14.6~kg)}{(2)(6.37\times 10^6~m+6.37\times 10^6~m)} - \frac{(6.67\times 10^{-11}~N~m^2/kg^2)(5.98\times 10^{24}~kg)(14.6~kg)}{(2)(6.37\times 10^6~m+1.911\times 10^7~m)}$ $E_B-E_A = 2.28\times 10^8~J - 1.14\times 10^8~J$ $E_B-E_A = 1.14\times 10^8~J$ Satellite B has a greater total energy than satellite A by $~~1.14\times 10^8~J$
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