# 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 | + | :'''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. |