Gravitational potential energy (GPE) is given by the formula:
\[ \text{GPE} = mgh \]
where:
- \( m \) is the mass (weight) of the object,
- \( g \) is the acceleration due to gravity, and
- \( h \) is the height climbed.
Now let's analyze the four options by calculating their potential energy changes.
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Carrying twice the weight and climbing half as high: \[ \text{GPE} = 2m \cdot g \cdot \frac{h}{2} = mg \cdot h \]
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Carrying half the weight and climbing half as high: \[ \text{GPE} = \frac{m}{2} \cdot g \cdot \frac{h}{2} = \frac{mg \cdot h}{4} \]
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Carrying twice the weight and climbing twice as high: \[ \text{GPE} = 2m \cdot g \cdot 2h = 4mg \cdot h \]
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Carrying half the weight and climbing twice as high: \[ \text{GPE} = \frac{m}{2} \cdot g \cdot 2h = mg \cdot h \]
Now compare the results:
- Option 1: \( mg \cdot h \)
- Option 2: \( \frac{mg \cdot h}{4} \)
- Option 3: \( 4mg \cdot h \)
- Option 4: \( mg \cdot h \)
The largest increase in gravitational potential energy comes from Option 3: carrying twice the weight and climbing twice as high, resulting in an increase of \( 4mg \cdot h \).
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