Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (4th Edition)

Published by Pearson
ISBN 10: 0133942651
ISBN 13: 978-0-13394-265-1

Chapter 13 - Newton's Theory of Gravity - Exercises and Problems: 45

Answer

The astronaut has 1.63 seconds to fire the rockets.

Work Step by Step

We can find the orbital speed for an object that is 400 km above the earth's surface. Note that the distance from the center of the earth is 6380 km + 400 km which is 6780 km. $v = \sqrt{\frac{G~M_e}{R}}$ $v = \sqrt{\frac{(6.67\times 10^{-11}~m^3/kg~s^2)(5.98\times 10^{24}~kg)}{6.78\times 10^6~m}}$ $v = 7.67\times 10^3~m/s$ $v = 7.67~km/s$ Since both objects are moving at a speed of 7.67 km/s but in opposite directions, the relative speed is 15.34 km/s. We can find the time it takes the two objects to cover a distance of 25 km. $t = \frac{distance}{speed}$ $t = \frac{25~km}{15.34~km/s}$ $t = 1.63~s$ The astronaut has 1.63 seconds to fire the rockets.
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