Here’s an amazing video in which a long row of stone blocks is arranged on-edge on top of a stone wall. Spoiler alert! I recommend watching the video before reading further!
Here’s what happens: Somebody pushes the leftmost block over, which knocks down the one to its right, which knocks down the one to its right, and so on in a chain reaction. This is more or less what you’d expect, just like a row of dominoes.
We notice that once the blocks fall, they don’t fall all the way down; the top of a block ends up resting on the bottom of the next block, overlapping a bit. That’s also not too surprising.
What is surprising is what happens once all the blocks fall. Almost instantly comes the amazing second chain reaction. Starting from the right, all the blocks that were resting on each other fall down flat, with virtually no space between them, making a perfect row of blocks laid end-to-end.
When students normally are exposed to pseudo-real-world problems, often called “word problems,” they are normally given exactly the information needed to solve it. But in the actual real world, we encounter actual problems and are given no hints about what facts are relevant. When I first saw this video I just wondered what was going on. I was flummoxed at first but gradually realized what’s happening. This is much closer to what really happens when we try to use mathematics to understand the world. But I’m not going to tell, at least not yet. You could do as I did, and just draw some pictures, make some assumptions, figure out what’s relevant, and figure it out. But to make it a little more, er, concrete, here are some starting points. Imagine the blocks are 12″ tall and 3″ wide when standing on edge (or 3″ high and 12″ long when lying flat). Here are some questions to think about:
- How far apart are the blocks initially? (Say, measured from the left side of one to the left side of the next.)
- Why don’t they fall to horizontal during the first chain reaction?
- Why do they all fall to horizontal after the rightmost one falls and starts the second chain reaction?
And, for extra credit, a few more that require trigonometry:
- What is the angle of a falling block as it hits the one to its right?
- What is the eventual resting angle of the blocks before the final chain reaction flattens them all?