Problem — Estimating the energy to climb stairs

Problem

Estimate how much energy you spend climbing five flights of stairs on foot (one storey $\sim 3\;\mathrm{m}$). Also express the result in kilocalories ($1\;\mathrm{kcal} \approx 4{,}2\;\mathrm{kJ}$). Compare with a snack ($\sim 100\;\mathrm{kcal}$). Is climbing the stairs enough to burn it off?

Solution

Estimated data. Mass of a person: $m \approx 70\;\mathrm{kg}$. Height gain: five storeys of $3\;\mathrm{m}$, i.e. $h \approx 5\cdot 3 = 15\;\mathrm{m}$.

Mechanical work (potential energy gained). $E_\text{mecc} = m\,g\,h = 70\;\mathrm{kg}\cdot 9{,}8\;\mathrm{m/s^2}\cdot 15\;\mathrm{m} \approx 1{,}0\cdot 10^4\;\mathrm{J}$

Metabolic energy. Muscles have an efficiency of about $25\%$: for every “useful” joule you burn about $4$. So the energy actually spent by the body is $E \approx \frac{E_\text{mecc}}{0{,}25} \approx 4\cdot 10^4\;\mathrm{J}$ $\ev{E \sim 4\cdot 10^4\;\mathrm{J}}$

In kilocalories. $E \approx \frac{4\cdot 10^4\;\mathrm{J}}{4{,}2\cdot 10^3\;\mathrm{J/kcal}} \approx 10\;\mathrm{kcal}$

Comparison. A snack provides $\sim 100\;\mathrm{kcal}$, i.e. about ten times the energy of this climb. The stairs alone are not enough: to burn off the snack you would need to climb those five flights about ten times.

Collegamenti

Argomenti: Lavoro ed energia Concetti: Energia potenziale gravitazionale Competenze: Stime di Fermi