What did the rocket boys prove in October Sky?

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The boys in October Sky proved many things, especially in regards to their own abilities, but there is one instance in particular where they make an especially critical proof. They are barred from creating any more rockets, and Homer is sent to work in the coal mines where his father works when a nearby forest catches fire and burns down due to a small rocket test. It is assumed, because of the public nature of their rocketry and their amateurism, that they caused the fire. However, using advanced differential calculus Homer teaches himself while working at the mines, they mathematically prove that their rocket could not have landed at the origin location of the fire. They use this math to locate their rocket, which is nowhere near the fire, and to find the actual culprit and deduce that it was from a nearby government base. They are lauded as heroes and allowed to practice rocketry again.

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