(a)
The expression for the speed of each planet and for their relative speed.
(a)
Answer to Problem 57P
The expression for the speed of planet
Explanation of Solution
The system of two planets are isolated, so the energy as well as the momentum is conserved.
Write the expression for the conservation of energy.
Here,
Write the expression for
Here,
Write the expression for the
Here,
Use equation (II) and (III) in (I) to solve for the conservation of energy.
In the initial condition, the two planets are nearly at rest and they are infinite distance apart.
Use equation (V) in (IV) to solve for the conservation of energy.
Write the expression for the conservation of momentum.
Here,
Use equation (V) in (VII) to solve for the conservation momentum.
Write the expression for the kinetic energy.
Here,
Write the expression for the momentum.
Write the expression for the
Here,
Use equation (IX) and (XI) in (VI) and it becomes,
Use equation (X) in (VIII) to solve for
Use equation (XIII) in (XII) to solve for
Substitute the value of
Use equation (XV) to solve for
Use equation (XVI) in (XIII) to solve for
Write the expression for the relative velocity.
Here,
Use equation (XVI) and (XVII) in (XVIII) to solve for
Conclusion:
Therefore, the expression for the speed of planet
(b)
The kinetic energy of the each planet just before the collision.
(b)
Answer to Problem 57P
The kinetic energy of the planet
Explanation of Solution
Write the expression for the
Here,
Use equation (XVI) in (IX) to solve for
Here,
Use equation (XVII) in (IX) to solve for
Here,
Conclusion:
Substitute
Substitute
Substitute
Therefore, the kinetic energy of the planet
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Chapter 11 Solutions
Principles of Physics: A Calculus-Based Text
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