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From Wikipedia's energy efficiency in transport, I gathered the following data for a person riding a bike:

65 kg person riding at 4.5 m/s requires 110 kJ/km, which equates to a continuous power output of 495 W. I assumed the mass of the bike was 10 kg, so total of 75 kg.

I want to convert these numbers into energy per ton-km, but I'm getting confused with something. First I start with 110 kJ/75kg-km. So now I want ton-km.

Dimensional analysis would seem to require multiplying the denominator by 1000 kg / 75 kg = 13.333... So then I get 110/13.333... = 8.25 kJ/ton-km.

But that cannot be right. It's gone the wrong direction. Spending 110 kJ over a km of distance with 75 kg weight cannot possibly equate to spending 13 times less for 1 ton of weight over the same distance. It can't be right. It should be 13 times more, not less.

What am I doing wrong? Why isn't dimensional analysis working here? What am I missing?

Maybe someone can help explain this kJ/ton-km business too. The problem is, it involves three units, not two. I'm not even sure if you can divide normally there. E.g., what if you have something like 50 kJ/ 20kg-600m ? Do you divide by both 20 and 600? Is that valid?

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2 Answers 2

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You just need to write out the dimensional analysis correctly. \begin{equation}\frac{110 \mathrm{kJ}}{75\mathrm{kg-km}} \times \frac{1000 \mathrm{kg}}{1 \mathrm{t}}=1466.66\mathrm{\frac{kJ}{t-km}} \end{equation}

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To keep a fraction value the same, you need to multiply (or divide) both the numerator and denominator by the same value. It's true, even if the numbers have units.

So:

It takes$$110 \text{ kJ to move }75 \text{ kg 1 km }$$ $$\frac{110}{75} \text{ kJ to move } \frac{75}{75} \text{ kg 1 km }$$ $$\frac{110 \times 1000}{75} \text{ kJ to move } \frac{75\times 1000}{75} \text{ kg 1 km }$$ $$1466.7 \text{ kJ to move } 1000 \text{ kg 1 km }$$

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