# Why do we need more power to do a job fast?

Let's assume we have two identical electric trains. One has a big electric motor (high power) and the other has small motor (low power). Let us assume the electric motors are of the same brand and the power of the motor is directly proportional to the size. Now, the two trains used their max power to travel between to stations and the powerful train arrives quicker. So can we relate time (the saved time) to the size (mass) of the motor or if we neglect the mass variance of the motors, to the mass of additional coal used at the power station? If this so, we can now measure time in kg's or lb's of coal?

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In physics, the average power $P$ is defined as the amount of work done per unit time interval, i.e., $P = \frac{\Delta W}{\Delta t}$. So, a greater $P$ implies a smaller $\Delta t$ for the same amount of work. This answers the first part of your question. As regards whether you can measure time in terms of kgs of coal, you could. Assume that the power output of the train engine scales linearly with the mass of the coal. You could define 3 minutes as the time it takes for the train to burn up, say, 100 kgs of coal. Then, any other time interval can be measured in terms of the mass of coal. So, 6 minutes is the time required to burn up 200 kgs of coal. Whereas this may be useful to the driver of the train, it is not a very useful definition to the rest of us because it is not at all clear what the efficiency of the engine is, the mass of the train is, how he accelerates and decelerates, etc.

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Just want to point out an inconsistency in your example: if it takes the train 3 minutes to burn up 100 kg of coal, it won't burn up 50 kg but 200 kg in 6 minutes. You might want to correct that. –  Wouter Feb 13 '13 at 15:03
Edited, thanks! –  contrariwise Feb 27 '13 at 21:50