Editing A bug in my high school physics intuition

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You're sitting on a bicycle or in a rocket ship, stationary. This gets boring, so you accelerate to 1 m/s. After cruising for a while you get bored again, so you accelerate some more, up to 2 m/s. What took more energy, getting from 0 to 1 m/s, or getting from 1 to 2 m/s?
 
You're sitting on a bicycle or in a rocket ship, stationary. This gets boring, so you accelerate to 1 m/s. After cruising for a while you get bored again, so you accelerate some more, up to 2 m/s. What took more energy, getting from 0 to 1 m/s, or getting from 1 to 2 m/s?
  
 
:'''Answer 1.''' Once you're cruising at 1 m/s, we may as well use the inertial reference frame which is moving at 1 m/s with you, in which you are stationary. Then the second spurt of acceleration corresponds to speeding up from 0 to 1 m/s, which is exactly what the first spurt did in the stationary reference frame. So the two take '''the same amount of energy'''.
 
:'''Answer 1.''' Once you're cruising at 1 m/s, we may as well use the inertial reference frame which is moving at 1 m/s with you, in which you are stationary. Then the second spurt of acceleration corresponds to speeding up from 0 to 1 m/s, which is exactly what the first spurt did in the stationary reference frame. So the two take '''the same amount of 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'''.
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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 $1^2 - 0^2 = 1$ unit of energy and the second one took $2^2 - 1^1 = 3$ 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.
 
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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