Energy comparison – potential energy of a raised body vs. kinetic energy of a falling body For the same mass and height reference, the potential energy of a vertically raised body is ______ the kinetic energy gained by a body after freely falling through that height (neglect air resistance).
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Aequal to
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Bless than
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Cgreater than
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Dcannot be compared without more data
Answer
Correct Answer: equal to
Explanation
Introduction / Context:This question examines conservation of mechanical energy for vertical motion under gravity. When air resistance is negligible, potential energy (PE) and kinetic energy (KE) interconvert.
Given Data / Assumptions:
- Mass m is the same in both comparisons.
- Height h is the same (reference taken from the lower level).
- Gravitational acceleration g is constant; air resistance is neglected.
Concept / Approach:Potential energy at height h is PE = m g h. If a body falls freely from rest through height h, its loss in potential energy equals its gain in kinetic energy by energy conservation, so KE at the lower level becomes m g h.
Step-by-Step Solution:
Compute PE at height h: PE = m * g * h.Consider free fall from rest: using energy conservation, initial total energy = PE at top = m g h, final total energy = KE at bottom (taking PE at bottom as zero).Hence KE_bottom = m g h.Therefore, PE at height h equals KE gained after falling through height h.Verification / Alternative check:Use kinematics: v^2 = u^2 + 2 g h with u = 0 ⇒ v^2 = 2 g h ⇒ KE = 0.5 m v^2 = 0.5 m (2 g h) = m g h, agreeing with energy conservation.
Why Other Options Are Wrong:
- Less than / greater than: contradict conservation of mechanical energy for ideal free fall.
- Cannot be compared: sufficient information is provided under standard assumptions.
Common Pitfalls:Forgetting to neglect air resistance; with drag, some energy is dissipated as heat and sound, so KE would be less than m g h.
Final Answer:equal to