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Why Thrown Objects follow Quadratic Paths (Tanton: Mathematics)

In algebra class one studies quadratic curves. Students are always told that they are relevant because the U-shaped paths objects thrown in the air follow are the same U-shapes that arise from quadratic curves. How do we know? Not all Us in nature are the same! Most books don't give an explanation as it involves calculus (just accept it as a fact and now do these 50 problems). But it is possible to give an explanation that seems mathematically right, and at the same time pinpoints what the real calculus issue is. Here it is!
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