So, this question arose when I was pondering the meaning of horsepower and torque in cars. I thought of the following question:

There is a 1 kilogram weight on planet M. Planet M has no atmosphere and has a gravitational constant of 1 meter per second per second. The weight is raised to a height of 2 meters and dropped. The weight will fall through the first meter in 1 second and the second meter in 0.4 seconds (rounding a bit).

So, in terms of work, gravity does 1 Joule of work on the weight over each 1 meter interval. However, in terms of power, gravity exerts 1 watt of power over the first interval and 2.5 watts over the second interval. Why does it seem that gravity is magically more powerful when the object is travelling faster?

The same question applies to a rocket ship in space. Assuming infinite fuel, the rocket will exert a constant force no matter the speed and burn fuel at the same rate, but from a constant reference frame, from where the rocket launched, it will appear that the faster the rocket travels the more energy it expends per second.

I think these questions accurately portray the confusion I've had with the concept of power ever since physics 2 in college so many years ago.

Thanks for reading.


The gravitational potential energy of the object is $mgz$, where $z$ is the height of the object above some reference point. So if the object moves downwards by one meter, it loses a fixed amount of energy, which in this particular example is one Joule. The potential energy is associated purely with its position --- wherever the object is, if you move it down by one meter, it will lose one Joule of energy.

Because the object covers the second meter more quickly than the first, it loses this fixed amount of energy more quickly over the second meter than the first. That's all there is to it.

To give an analogy with an elastic band: if you stretch an elastic band, its potential energy will be at some increased level. The potential energy depends on just how stretched it is --- the extension of the band from its equilibrium. When you let the band go, its potential energy is going to decrease. But as you let the band go, the contraction will speed up over time. You would have to watch it in slow-motion, but in the first millisecond the band wouldn't contract as much as in the second millisecond, because in the second millisecond the band will have 'got up to speed', hence will be contracting more quickly, hence will be losing potential energy faster.

I haven't said anything new at all here, but maybe from another perspective the matter seems less magical.

Problems like these often boil down to a clash of definition and intuition --- that is to say, your intuition of what energy is may not match its mathematical definition. I was once asked by a friend the question: how can the Earth just keep attracting things to itself indefinitely; won't it run out of energy? Giving a satisfying answer here is hard --- if we define energy in the appropriate way, it's clear that the Earth won't 'run out of it'.

Similarly, the fact that 'gravity gives the object more energy in the second second than the first' is just a consequence of our definition of energy. I mean, when you watch the object fall at constant acceleration, nothing about the motion seems odd or counter-intuitive. But if we look at the quantity

$$\frac{1}{2} m v^2 $$

and how it changes over time, we'll see that it increases more over the second second than the first. This is just a consequence of the way it's defined!

So gravity is more 'powerful' when the object is moving faster. But 'powerful' here has a precise meaning. Gravity exerts the same force on the object regardless of its speed --- if one were to think of 'power' as 'how much force it can exert' (which is NOT the physical definition) then gravity is just as 'powerful' no matter the speed of the object, which to you seems sensible and intuitive. Gravity only gives the object more energy over the second second than the first as a consequence of the definition of energy.

Hope this helps.

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  • $\begingroup$ Yes, it seems better to just live with the fact that the definition of power doesn't seem to satisfy intuition for applications such as gravity. For applications in cars and machines involving work done by contact it is slightly more intuitive. $\endgroup$ – Ian Jan 2 '14 at 20:52

Work is force times displacement. $$W = \vec{F} \cdot \vec{d}$$

Power is work over time. $$P = \frac{dW}{dt}$$

Therefore, for constant force, power is force times (displacement over time), that is, force times velocity. $$P = \frac{d}{dt} (\vec{F} \cdot \vec{d}) = \vec{F} \cdot \frac{d\vec{d}}{dt} = \vec{F} \cdot \vec{v}$$

It's not specific to gravity, and it's not magic. Just math.

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  • $\begingroup$ Yes, the mathematical equations are as you wrote them. But it still doesn't make that much sense. $\endgroup$ – Ian Dec 31 '13 at 19:17

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