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Scaling and Proportion Physics problem. Please Help!?
1.
A flea is able to jump straight up about 0.44 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 700, the cross section of its muscle would increase by 700^2 and the length of contraction would increase by 700. How high would this "superflea" be able to jump? (Don't forget that the mass of the "superflea" increases as well.)
My attempt:
The muscle cross section increases by 700^2 = 490000. The energy stored in the muscle increases by area x length = 700^3 = 343 000,000. The flea's mass increases by the cube of length, or the same ratio, 343,000,000.
m= 0.44^3
= 0.085184
M g H = stored muscle energy.
H = 343 000,000/ (0.085184)(9.8)
= 4.1 x 10^8
H, the jumping height, is proportional to
(stored muscle energy/M)
and this ratio does not change. Neither does the height that it can jump.
I AM REALLY CONFUSED! PLEASE HELP ME WITH MY METHOD BECAUSE I THINK IM DOING IT WRONG
1.
A flea is able to jump straight up about 0.44 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 700, the cross section of its muscle would increase by 700^2 and the length of contraction would increase by 700. How high would this "superflea" be able to jump? (Don't forget that the mass of the "superflea" increases as well.)
My attempt:
The muscle cross section increases by 700^2 = 490000. The energy stored in the muscle increases by area x length = 700^3 = 343 000,000. The flea's mass increases by the cube of length, or the same ratio, 343,000,000.
m= 0.44^3
= 0.085184
M g H = stored muscle energy.
H = 343 000,000/ (0.085184)(9.8)
= 4.1 x 10^8
H, the jumping height, is proportional to
(stored muscle energy/M)
and this ratio does not change. Neither does the height that it can jump.
I AM REALLY CONFUSED! PLEASE HELP ME WITH MY METHOD BECAUSE I THINK IM DOING IT WRONG
= (stored muscle energy/M)*gWhere H is the maximum jumping height, M is the mass of the flea, and g is the acceleration due to gravity (9.8 m/s2).In this problem, the mass of the flea increases by a factor of 343 000,000, while the stored energy increases by a factor of 7003.Plugging in these values into the equation, we get