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

[sam_hall88]

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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How do I get started on this problem?

 

[mgb_phys]

Here is a good start: http://fathom.lib.uchicago.edu/2/21701757/

 

[m2003]

To solve this problem, we need to use the principles of work, force, and energy. Let's break down the steps to find the maximum height that the "superflea" could jump:

1. Determine the mass of the "superflea": Since we are magnifying the flea by a factor of 500, its mass would also increase by 500^3, which is equal to 125,000,000. So the mass of the "superflea" would be 125 million times the mass of a regular flea.

2. Calculate the maximum force exerted by the "superflea's" muscle: As mentioned in the content, the maximum force exerted by a muscle is proportional to its cross-sectional area. So, if we increase the flea's cross-sectional area by 500^2, the maximum force would also increase by 500^2, which is equal to 250,000. Therefore, the maximum force exerted by the "superflea's" muscle would be 250,000 times the maximum force exerted by a regular flea's muscle.

3. Calculate the work done by the "superflea's" muscle: The work done by a muscle is equal to the force exerted by the muscle multiplied by the length of contraction. Since we have determined the maximum force exerted by the "superflea's" muscle, we now need to determine the length of contraction. According to the content, the length of contraction would also increase by a factor of 500. So, if a regular flea's muscle contracts 0.51 m, the "superflea's" muscle would contract 500 times that, which is equal to 255 m. Therefore, the work done by the "superflea's" muscle would be 250,000 times the work done by a regular flea's muscle.

4. Calculate the gravitational potential energy gained: When an animal jumps, the work done by its muscles is converted into gravitational potential energy. This energy is equal to the mass of the animal multiplied by the acceleration due to gravity (9.8 m/s^2) and the height it jumps. So, using the mass and work calculated in the previous steps, we can calculate the gravitational potential energy gained by the "superflea" as it jumps.

5. Calculate the height the "superflea" can jump: Now that we have all the necessary values, we can use the equation