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I've asked this once before, but it didn't word it well and didn't get the answer I was looking for, so here is my question:

Imagine you are in a vacuum and friction acting on the object you are pushing is negligible:

If you increase something's speed from 0 to 10 m/s it uses a certain amount of energy (e) and a certain amount of force (f) applied over t time. If you increase it's speed from 10 to 20 m/s it takes f force over t time again, but uses >1e energy since ke = 1/2mv^2. If the force is constant, then the stress on your arm pushing the object is constant, so how does the extra energy get into the object being pushed? If it transferred through your arm then why would the stresses on your arm not increase?

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    $\begingroup$ Presumably you imagine it will require more than $e$ energy to get from 10 to 20 m/s because of friction/air resistance, right? If there's a larger resistance but you're applying the same force then it will take more time to get from 10 to 20 m/s than to get from 0 to 10 m/s, in other words you're applying the same force but for longer, and so that's where the extra energy comes from. $\endgroup$ – lemon Jan 14 '18 at 23:35
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It does indeed take the same impulse to increase the speed from 10 to 20 as from 0 to 10. Impulse is force times time. It takes three times as much energy to increase the speed from 10 to 20 as from 0 to 10, because energy (work) is force times distance. When it's moving faster, it goes further during the time you're applying the force, so even though the force and time remain the same, the distance and therefore the energy required is greater.

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