One example is the story of Rapunzel. She was a girl whose hair was very long and beautiful, and grew very fast. Every time she cut her hair, it grew twice as fast.
She was imprisoned by a witch, who put her in a tower. The tower was as tall as twelve men. It had no stairs and no door, so the only way to get to Rapunzel was to call for her to let down her hair through the window, and then use the hair to climb up. Eventually a prince comes along, and uses this method to enter the tower.
There are many things about this that makes Bengt think.
"Is it possible to climb up someone's hair like that? It would be pretty uncomfortable for the girl... but I guess she could just wrap it a couple of times around something at the end near her head, before letting it out."
He assumes that all men, like himself, weigh about 100 kg, and are about 180 cm tall. He also assumes that Rapunzel is only 120 cm, and weighs 40 kg. This is clearly a rather short girl, but Bengt happens to like short girls.
Bengt pulls out one of his own hairs, to test its strength. It turns out it can carry the weight of ten grams, and can be extended by about 5% of its normal length.
Next, he wants to see how thick his hair is. He gets out a red laser pointer, and points it at the single hair. On the wall two meters away, he sees the diffraction pattern. The first black band is 15 mm from the center.
Bengt decides Rapunzel was probably blonde. His own hair is some sort of brownish colour. He happens to know that blonde people have about 50% more hairs, and they are about half as thick. On the other hand, they both have rather straight hair, which means that the cross section of the hair is circular.
Bengt realises he needs to do one more test. He immerses his hair in water, and somehow manages to determine that the volume of the displaced water is 0.8 ml. He also decides that "displace" is a funny word, and makes a mental note to start using it instead of "misplace", if only to annoy people.
He dries his hair, happy that it is only a decimeter long, so no one will be climbing in it any time soon.
- If Bengt was to cut his hair only when it reached the floor, and then cut off all of it, he would have to cut his hair once every 15 years. If Rapunzel, with her ever-increasing hair growth speed, did the same, what would eventually happen?
- How long could she keep doing that? (Assume that she starts off with the same growth speed as Bengt.)
- Suppose instead that her hair grows at normal speed. How long would it take for the hair to reach the ground outside the tower?
- Sometimes the prince might need to make a hasty exit. He can then grab a hold of the hair, and jump out through the window. Assuming that he can hold on to the hair, for how tall towers would this be possible?
- Without any jumping, how many princes could hang on Rapunzel's hair at once?
- For best result, should they be spaced equally, or all hanging at the end?
Posts
