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My teacher gave us a worksheet with word problems and their solutions. It is in German, so I have tried my best to translate it to English:

A 26 year old man climbs Mount Everest (8848 m) in only 8 hours 10 mins from the base camp at 5300 m. Estimate the "lifting work" (Hubarbeit) that the man exerted in the climb.

I thought to use the formula: $W = F \cdot \Delta S$, but I don't know what $F$ would be. I think the $\Delta S$ is $8848 - 5300 = 3548$. But then I looked at the solutions and my teacher used this formula: $W = m g h$. He guessed the man's weight and also used $g = 10\ \mathrm{N}/\mathrm{kg}$. I don't understand this.

Could someone maybe help me? Anyone have an idea?

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"Estimate the 'lifting work'" is the key part. Well, what is the lifting work? What are you lifting against? Gravity. What's the force gravity exerts? $F_{gravity} = mg$ (neglecting variations in $g$ as you go up the mountain). Therefore, the force in your formula $W=F\cdot \Delta S$ is the force of gravity.

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$W=mgh$ is the potential energy the climber (of mass $m$) gains by changing his altitude by $h=8848~\textrm{m} - 5300~\textrm{m} = 3548~\textrm{m}$.

This will be a lower bound for the work done by the climber. In reality work/energy will be lost in friction and any other activities (e.g. setting up a tent...) the climber does which are not related to raising his altitude.

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He's asking for "lifting work", work in the vertical direction. If we assume he's not accelerating up Everest, his vertical work will be opposing gravity, so $$F_{Lifting} = mg$$ and $$W_{Lifting} = F_{Lifting} \Delta h = (mg)\Delta h$$

which is what your teacher did.

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