Potential Energy (gravity & spring)

[sebmista]

alright guys I've been going at this problem for a while I can't seem to figure it out. Here it is

A flea is able to jump straight up about 0.40 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 550, the cross section of its muscle would increase by 550^2 and the length of contraction would increase by 550. How high would this "superflea" be able to jump? (Don't forget that the mass of the "superflea" increases as well.)

So this is the set up I've come up with...
The initial gravity PE is 0 and the Initial and Final Kinetic energies are 0 as well... The final spring energy is 0 too so I'm left with
1/2kx^2 initial = mgy final y = .40 but I'm left with 3 other unknowns... If i knew them I could just multiply them all by 550 right?
HELP

 

[OlderDan]

The values of the unknowns are not really important. Make some reasonable assumptions for the mass of the flea and the area and contraction lengths of the muscles and see if you can do the problem.

 

[sebmista]

Hmmm... Ok so then I do the following estimations...

m = .000001 kg
area = 1 millimeters (10^-3)
contraction = .000001 micrometer (10^-6)

so then it would be
m = .000001 x 550 = .5.5e-4 kg
area = .001 x 550^2 = 302.5 meters ? this seems weird...
contraction = .000001 x 550 = 5.5e-4 meters

Am I going in the right direction?

 

[OlderDan]

Hmmm... Ok so then I do the following estimations...

m = .000001 kg
area = 1 millimeters (10^-3)
contraction = .000001 micrometer (10^-6)

so then it would be
m = .000001 x 550 = .5.5e-4 kg
area = .001 x 550^2 = 302.5 meters ? this seems weird...
contraction = .000001 x 550 = 5.5e-4 meters

Am I going in the right direction?

area would be square millimeters, not just milimeters, but the numerical part is OK

Since the numbers do not reallly matter, you could use simpler ones. It will be best if you express things in mks units. The mass in kg is good. The area should be some number of meters squared. If you square your 1mm that would be .0000001 m^2. Your contraction number is OK if expressed in meters, but your number is incredibly small. 1mm or .001m would be fine and easier to keep track of.

The important thing is the information that work done in contracting the muscle is the same as the gravitational potential energy achieved when jumping. From the information given that work is proportional to the are times the length of the muscle. Can you write an equation that expresses that relationship? And what happens to the mass of the flea if all the lengths are multiplied by 550?