Editing A bug in my high school physics intuition

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:'''Answer 2.''' Just measure the kinetic energy differences. Kinetic energy is proportional to the square of speed, so the first spurt of acceleration took 1<sup>2</sup> - 0<sup>2</sup> = 1 unit of energy and the second one took 2<sup>2</sup> - 1<sup>2</sup> = 3 units of energy, which is '''3 times as much energy'''.
 
:'''Answer 2.''' Just measure the kinetic energy differences. Kinetic energy is proportional to the square of speed, so the first spurt of acceleration took 1<sup>2</sup> - 0<sup>2</sup> = 1 unit of energy and the second one took 2<sup>2</sup> - 1<sup>2</sup> = 3 units of energy, which is '''3 times as much energy'''.
  
The problem is that both of these answers seem intuitively compelling. For a similar problem, I thought of one answer but not the other, and it seemed so clear that it wasn't even worth double checking. After I realized what had happened, I started on a quest to update my intuition so that the same failure mode doesn't happen again. I think I've at least partially succeeded. In any case, I was at least able to console myself that many people with very good physical intuition also found this confusing. Let's review some possible explanations.
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The problem is that both of these answers seem intuitively compelling. For a similar problem, I thought of one answer but not the other, and it seemed so clear that it wasn't even worth double checking. After I realized what had happened, I started on a quest to update my intuition so that the same failure mode doesn't happen again. I haven't entirely succeeded, but I was at least able to console myself that many people with very good physical intuition also found this confusing. Let's review some possible explanations.
  
  
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As expected, the work depends on more than the impulse. It also depends on the ''difference'' between their speeds and on how much their speeds change (or equivalently, on their masses). Those things are agreed on by people in different reference frames (let's ignore relativistic speeds for now). So even though the amount of work you do ''on a given object'' depends on the reference frame, the total work done by exerting a fixed amount of impulse is independent of reference frame. Whew! It [[#"It depends on your reference frame"|would have been crazy if that weren't true]].
 
As expected, the work depends on more than the impulse. It also depends on the ''difference'' between their speeds and on how much their speeds change (or equivalently, on their masses). Those things are agreed on by people in different reference frames (let's ignore relativistic speeds for now). So even though the amount of work you do ''on a given object'' depends on the reference frame, the total work done by exerting a fixed amount of impulse is independent of reference frame. Whew! It [[#"It depends on your reference frame"|would have been crazy if that weren't true]].
 
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