Physics For Scientists And Engineers With Modern Physics, 9th Edition, The Ohio State University
Physics For Scientists And Engineers With Modern Physics, 9th Edition, The Ohio State University
9th Edition
ISBN: 9781305372337
Author: Raymond A. Serway | John W. Jewett
Publisher: Cengage Learning
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Chapter 13, Problem 61AP

(a)

To determine

The magnitude of the speed of each planets and relative speed.

(a)

Expert Solution
Check Mark

Answer to Problem 61AP

The magnitude of speed of each planet respectively m22Gd(m1+m2), m12Gd(m1+m2) and their relative velocity is 2G(m1+m2)d.

Explanation of Solution

Initially both planets will have zero potential and kinetic energy.

Write expression for the conservation of energy.

  0=12m1v12+12m2v22Gm1m2d                                                                                  (I)

Here, v1 is the speed of the planet with mass m1, v2 is the speed of the planet with mass m2.

The initial momentum of the both planet is zero.

Write the expression for the conservation of momentum.

  0=m1v1m2v2

Rewrite the above equation in terms of v2.

  m2v2=m1v1v2=m1v1m2                                                                                                              (II)

Write the expression to calculate the relative velocity of the two planets.

  V=v1(v2)                                                                                                         (III)

Here, V is the relative velocity of the planets.

Conclusion:

Substitute the equation (II) in (I) to rewrite in terms of v1.

  12m1v12+12m2(m1v1m2)2=Gm1m2dv12+(m1m2)v12=2Gm2dv12(1+m1m2)=2Gm2dv1=2Gm22d(m1+m2)v1=m22Gd(m1+m2)

Use the above expression in the equation (II) to rewrite in terms of

  v2=m1(m22Gd(m1+m2))m2=(m12Gd(m1+m2))

Substitute the expression for v1 and v2 in the equation (III) to calculate V.

  V=(m22Gd(m1+m2))((m12Gd(m1+m2)))=(m1+m2)2Gd(m1+m2)=2G(m1+m2)d

Therefore, the magnitude of speed of each planet respectively m22Gd(m1+m2), m12Gd(m1+m2) and their relative velocity is 2G(m1+m2)d.

(b)

To determine

The kinetic energy of each planet.

(b)

Expert Solution
Check Mark

Answer to Problem 61AP

The kinetic energy of each planet is respectively 1.06×1032J and 2.66×1031J.

Explanation of Solution

Write the expression to calculate relative distance between the two planets before collision.

  d=r1+r2                                                                                                                (IV)

Write the expression to calculate the velocity of the planet of mass m1.

  v1=m22Gd(m1+m2)                                                                                             (V)

Write the expression to calculate the velocity of the planet of mass m2.

  v2=m12Gd(m1+m2)                                                                                              (VI)

Write the expression to calculate the kinetic energy of the planet of mass m1.

  K1=12m1v12                                                                                                          (VII)

Here, K1` is the kinetic energy of the planet with mass m1.

Write the expression to calculate the kinetic energy of the planet with mass m2.

  K2=12m2v22                                                                                                  (VIII)

Here, K2 is the kinetic energy of the planet with mass m2.

Conclusion:

Substitute 3.00×106m for r1 and 5.00×106m for r2 in the equation (IV) to calculate d.

  d=3.00×106m+5.00×106m=8.00×106m

Substitute 8.00×106m for d, 6.67×1011Nm2/kg2 for G, 2.00×1024kg for m1 and 8.00×1024kg for m2 in the above equation to calculate v1.

  v1=8.00×1024kg2(6.67×1011Nm2/kg2)8.00×106m(2.00×1024kg+8.00×1024kg)=1.03×104m/s

Substitute 8.00×106m for d, 6.67×1011Nm2/kg2 for G, 2.00×1024kg for m1 and 8.00×1024kg for m2 in the above equation to calculate v2.

  v2=2.00×1024kg2(6.67×1011Nm2/kg2)8.00×106m(2.00×1024kg+8.00×1024kg)=2.58×103m/s

Substitute 2.00×1024kg for m1 and 1.03×104m/s for v1 in the equation (VII) to calculate K1.

  K1=12(2.00×1024kg)(1.03×104m/s)2=1.06×1032J

Substitute 8.00×1024kg for m2 and 2.58×103m/s for v2 in the equation (VIII) to calculate K2.

  K2=12(8.00×1024kg)(2.58×103m/s)2=2.66×1031J

Therefore, the kinetic energy of each planet is respectively 1.06×1032J and 2.66×1031J.

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Chapter 13 Solutions

Physics For Scientists And Engineers With Modern Physics, 9th Edition, The Ohio State University

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