Ch 13: Newton's Theory of Gravity

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 13: Newton's Theory of Gravity

Problem 67

Chapter 13, Problem 67

Two Jupiter-size planets are released from rest 1.0 x 10¹¹ m apart. What are their speeds as they crash together?

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Identify the key concepts involved: This problem involves gravitational potential energy and kinetic energy. The planets start from rest, so their initial kinetic energy is zero. As they move toward each other due to gravitational attraction, potential energy is converted into kinetic energy.

Write the expression for the total mechanical energy of the system. The total energy is conserved and can be written as: \( E = U + K \), where \( U \) is the gravitational potential energy and \( K \) is the kinetic energy. Initially, \( K = 0 \), so \( E = U_{initial} \).

Express the gravitational potential energy: \( U = -\frac{G m_1 m_2}{r} \), where \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses of the two planets, and \( r \) is the distance between them. Initially, \( r = 1.0 \times 10^{11} \; \text{m} \).

At the moment of collision, the distance between the planets becomes zero (or effectively their radii if they are not point masses). The final kinetic energy of the system is equal to the decrease in gravitational potential energy. Use the conservation of energy: \( U_{initial} = K_{final} + U_{final} \). Solve for \( K_{final} \), which is the total kinetic energy of the system.

Relate the kinetic energy to the speed of the planets: The kinetic energy of each planet is \( K = \frac{1}{2} m v^2 \). Use the fact that the planets have equal masses and move symmetrically to find their individual speeds. Solve for \( v \) using the total kinetic energy and the mass of one planet.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Gravitational Force

The gravitational force is the attractive force between two masses, described by Newton's law of universal gravitation. It states that the force is directly proportional to the product of the masses and inversely proportional to the square of the distance between their centers. This force is crucial in determining how the planets will accelerate towards each other as they move closer.

Conservation of Energy

The principle of conservation of energy states that the total energy in a closed system remains constant. In this scenario, the gravitational potential energy of the two planets will convert into kinetic energy as they move towards each other. Understanding this concept allows us to calculate the speeds of the planets just before they collide.

Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. In this problem, kinematic equations can be used to relate the distance traveled by the planets to their speeds as they approach each other. This is essential for determining their final velocities at the moment of collision.

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