A few thoughts about my own experience with the slinky problem.
I recall that it took me something like an hour to figure out that the video is not a trick and what the underlying physics was. I could be wrong on the time frame, though, as it was about a year ago.
I did not write any differential equations, however, just had to make the connection between the fact that the bottom was not moving and the conclusion that the top was essentially a shock wave. After that all that was left is to verify that the longitudinal slinky waves (not sound waves) are indeed slow enough for the shock to form very quickly.
Now, had I read the Polya book and applied the mantra, would I have found the solution any faster? I doubt that, though there is no way to check. I was quite familiar with shock waves already, as well as with the pole-in-the-barn paradox, having had to explain the latter on IRC many times. However, I did not pay attention to the fact that the contraction of the pole after its front hits the gate is due to a shock wave, not a sound wave, the latter being much too slow in relativistic circumstances. Why did I miss that? Probably because it was not essential to resolving the paradox, as once you realize that there are no rigid bodies in relativity, the paradox goes away.
I suspect that I internalized the “have you seen a similar problem before?” approach as much as I could already, but not expecting to see shock waves on such a slow time scale delayed the realization significantly.
A few thoughts about my own experience with the slinky problem.
I recall that it took me something like an hour to figure out that the video is not a trick and what the underlying physics was. I could be wrong on the time frame, though, as it was about a year ago.
I did not write any differential equations, however, just had to make the connection between the fact that the bottom was not moving and the conclusion that the top was essentially a shock wave. After that all that was left is to verify that the longitudinal slinky waves (not sound waves) are indeed slow enough for the shock to form very quickly.
Now, had I read the Polya book and applied the mantra, would I have found the solution any faster? I doubt that, though there is no way to check. I was quite familiar with shock waves already, as well as with the pole-in-the-barn paradox, having had to explain the latter on IRC many times. However, I did not pay attention to the fact that the contraction of the pole after its front hits the gate is due to a shock wave, not a sound wave, the latter being much too slow in relativistic circumstances. Why did I miss that? Probably because it was not essential to resolving the paradox, as once you realize that there are no rigid bodies in relativity, the paradox goes away.
I suspect that I internalized the “have you seen a similar problem before?” approach as much as I could already, but not expecting to see shock waves on such a slow time scale delayed the realization significantly.