Question

Human Energy vs. Insect Energy. For its size, the common flea is one of the most accomplished jumpers in the animal world. A 2.0-mm-long, 0.50-mg flea can reach a height of 20 cm in a single leap. (a) Ignoring air drag, what is the take- off speed of such a flea? (b) Calculate the kinetic energy of this flea at takeoff and its kinetic energy per kilogram of mass. (c) If a 65-kg, 2.0-m-tall human could jump to the same height compared with his length as the flea jumps compared with its length, how high could the human jump, and what takeoff speed would the man need? (d) Most humans can jump no more than 60 cm from a crouched start. What is the kinetic energy per kilogram of mass at takeoff for such a 65-kg person? (e) Where does the flea store the energy that allows it to make sudden leaps?

Answer 1

(a) 1.98 m/s

Due to the law of conservation of energy, the kinetic energy of the flea at take off is equal to the gravitational potential energy of the flea at its maximum height:

`K=U\\(1)/(2)mv^2 = mgh`

where

`m=0.50 mg=0.5\cdot 10^(-3) kg` is the mass of the flea

v = ? is the take-off speed of the flea

g = 9.8 m/s^2 is the acceleration due to gravity

`h = 20 cm = 0.2 m` is the maximum height reached by the flea

Solving the formula for v, we find

`v=√(2gh)=√(2(9.8 m/s^2)(0.20 m))=1.98 m/s`

(b)
`9.8\cdot 10^(-4)J`, 1.96 J/kg

The kinetic energy of the flea at take-off is equal to:

`K=(1)/(2)mv^2=(1)/(2)(0.5\cdot 10^(-3)kg)(1.98 m/s)^2=9.8\cdot 10^(-4)J`

And the kinetic energy per kilogram of mass is

`(K)/(m)=(9.8\cdot 10^(-4) J)/(0.5\cdot 10^(-3) kg)=1.96 J/kg`

(c) 200 m, 62.6 m/s

The flea, which is 2.0 mm = 0.002 m long, can reach a height of 0.20 m. The man, which is 2.0 m tall, could reach a height given by the proportion:

`2.0 m : h= 0.002 m : 0.20 m\\h=((2.0 m)(0.20 m))/(0.002 m)=200 m`

Then his take-off speed would be given by the same formula used in step a):

`v=√(2gh)=√(2(9.8 m/s^2)(200 m))=62.6 m/s`

(d) 5.78 J/kg

The height the human can jump is h = 60 cm = 0.60 m. In this case, the take off speed is

`v=√(2gh)=√(2(9.8 m/s^2)(0.60 m))=3.4 m/s`

So the kinetic energy per kilogram of a m=65 kg person is given by

`(K)/(m)=(1)/(2m)mv^2=(1)/(2)v^2=(1)/(2)(3.4 m/s)^2=5.78 J/kg`

(e) In the elastic energy of the muscles

The energy that allows the flea (but also the human) to make the jump is initially stored as elastic energy of the muscles. During the jump, muscles release, converting this elastic energy into kinetic energy first, and later into gravitational potential energy.