# A bug in my high school physics intuition

I think of myself as having pretty good physical intuition, at least for plain old mechanics. Some systems are hard to get a grip on (e.g. why does a hard-boiled egg stand up when you spin it?), but at least I recognize when I'm confused about them. However, I recently got some simple mechanics really wrong, and didn't feel the slightest bit of confusion about it. So I went on a quest to update my intuition so that the same failure mode never happens again.

Here's the motivating question.

You're sitting on a bicycle or in a rocket ship, stationary. This gets boring after a while, 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 m/s to 1 m/s, or getting from 1 m/s to 2 m/s?

The problem is that there are two intuitively compelling answers, which severely conflict.

**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, 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 $1^2 - 0^2 = 1$ unit of energy and the second one took $2^2 - 1^1 = 3$ units of energy, which is**3 times as much energy**.