Question

A flea is able to jump straight up about 0.52 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 1040, the cross section of its muscle would increase by 1040² and the length of contraction would increase by 1040. How high would this "super flea" be able to jump? (Don't forget that the mass of the "super flea" increases as well.)

in meters

Answer 1

Using energy conservation for normal flea:

KEi + PEi + W = KEf + PEf

KEi = 0, since initially flea was at rest

PEi = 0, since flea was at ground

KEf = 0, since at the max height velocity of flea is zero

PEf = m*g*H_max

H_max = max height achieved by normal flea = 0.52 m

W = Work-done by the muscle = Fm*d

Fm = Force exerted by muscle

d = length of contraction

So,

0 + 0 + Fm*d = 0 + m*g*H_max

H_max = Fm*d/(m*g) = 0.52 m

Now when given 'Superflea' jumps then given that

flea is magnified by a factor of 1040, and the cross-section area of muscle increases by a factor of 1040^2 (It should be 1040^2 in the question as 1040 does not makes any sense, So check and let me know if 10402 is correct value)

length of contraction of flea is also increased by a factor of 1040, So d1 = 1040*d

Now we know that mass of Superflea also increase, So

Mass = Volume*density

Since density of any animal/object is always constant, And Volume = Cross-sectional Area*length of contraction, So

Mass = density*Cross-sectional Area*length of contraction

Now since given that cross-section area of muscle increases by a factor of 1040^2 and length of contraction of flea is also increased by a factor of 1040, So

Mass of superflea will also increase by a factor of 1040^2*1040 = 1040^3 factor

Fm1 = New force exerted by muscle = 1040^2*Fm, since given that maximum force exerted by a muscle is proportional to its cross-sectional area and cross section of superflea's muscle would increase by 1040^2, So

Now new max height which superflea can achieve will be:

H1_max = Fm1*d1/(m1*g)

H1_max = (1040^2*Fm)*(1040*d)/(1040^3*m*g) = 1040^3*Fm*d/(1040^3*m*g)

H1_max = Fm*d/(m*g) = H_max = 0.52 m

So Superflea can also be able to jump at height of 0.52 m only.

Let me know if you've any query.