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: 23

Answer

The comet's speed at the midpoint is 32.6 km/s

Work Step by Step

We can use conservation of energy to find the comet's speed when it is at the midpoint between the stars. When the comet is far away, we can assume that the kinetic energy is zero and the gravitational potential energy is zero. Note that $U_2$ is equal to twice the value of the gravitational potential energy between the comet and one of the two stars. Let $M_s$ be mass of each star and let $M_c$ be the comet's mass. Let $R$ be the distance from the comet to each star when the comet is at the midpoint. $K_2+U_2 = K_1+U_1$ $\frac{1}{2}M_c~v^2 = 0+0-(-\frac{2~G~M_s~M_c}{R})$ $v^2 = \frac{4~G~M_s}{R}$ $v = \sqrt{\frac{4~G~M_s}{R}}$ $v = \sqrt{\frac{(4)(6.67\times 10^{-11}~m^3/kg~s^2)(1.99\times 10^{30}~kg)}{5.0\times 10^{11}~m}}$ $v = 3.26\times 10^4~m/s$ $v = 32.6~km/s$ The comet's speed at the midpoint is 32.6 km/s.
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