A satellite revolves around the Earth in a circular orbit with a mechanical energy of -10^12 J. What is its kinetic energy?
Understand the Problem
The question is asking for the kinetic energy of a satellite in a circular orbit around the Earth, given its mechanical energy. To solve this, we can use the relationship between total mechanical energy (E) and kinetic energy (K), where E = K + U, and for circular orbits, E is typically negative and U (potential energy) is related to the kinetic energy.
Answer
The kinetic energy of the satellite is $K = +10 \times 10^{11} \, J$.
Answer for screen readers
The kinetic energy of the satellite is: $$ K = +10 \times 10^{11} , J $$
Steps to Solve
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Understand the relationship between mechanical energy and kinetic energy Mechanical energy ( E ) is the sum of kinetic energy ( K ) and potential energy ( U ): $$ E = K + U $$
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Identify the relation between potential and kinetic energy in circular orbit For a satellite in circular orbit around a planet, the potential energy ( U ) is related to kinetic energy ( K ) by the equation: $$ U = -2K $$
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Substituting the relation into the mechanical energy equation Substituting ( U ) into the mechanical energy equation gives: $$ E = K - 2K = -K $$
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Solve for kinetic energy Since we know the mechanical energy ( E ): $$ K = -E $$ Substituting the given value of ( E = -10^{12} , J ): $$ K = -(-10^{12}) = 10^{12} , J $$
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Selecting the correct answer Since the calculated kinetic energy is ( 10^{12} , J ), we need to verify the units to match the answer choices. We express it in the scientific notation format: $$ K = +10 \times 10^{11} , J $$
The kinetic energy of the satellite is: $$ K = +10 \times 10^{11} , J $$
More Information
The kinetic energy of an object in circular orbit is significant for understanding how satellites and celestial bodies interact with gravitational forces. In this case, the negative mechanical energy indicates a bound system, which means the satellite remains in orbit due to its kinetic energy overcoming gravitational pull.
Tips
- Mixing signs: Ensure that when dealing with potential and mechanical energy, the signs are correctly handled, especially since potential energy is negative in bound systems.
- Misapplying formulas: It's essential to use the correct relationships between potential and kinetic energy specifically for circular orbits.
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