Solving the Superflea Jump Problem: How High Can it Jump?

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A flea can jump approximately 0.51 meters, and if scaled to human size, it could theoretically leap over a 100-story building. The mechanics of jumping involve converting muscle work into gravitational potential energy, with muscle force being proportional to its cross-sectional area. When magnifying a flea by 500 times, both the muscle cross-section and contraction length increase significantly, impacting jump height. However, the mass of the "superflea" also increases, complicating the calculations. Understanding these principles is essential for solving the superflea jump problem.
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A flea is able to jump straight up about 0.51 m. It has been said that if a flea were as big as a human, it would be able to jump over a 100 story building! When an animal jumps, it converts work done in contracting muscles into gravitational potential energy (with some steps in between). The maximum force exerted by a muscle is proportional to its cross-sectional area, and the work done by the muscle is this force times the length of contraction. If we magnified a flea by a factor of 500, the cross section of its muscle would increase by 500^2 and the length of contraction would increase by 500. How high would this "superflea" be able to jump? (Don't forget that the mass of the "superflea" increases as well.)
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