Question

BIO 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 takeoff 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 initial kinetic energy of the flea will be equal to its gravitational potential energy when it reaches the highest position in the jump:

`K_i = U_f\\(1)/(2)mv^2 = mgh`

where

`m=0.50 mg = 5\cdot 10^(-4)kg` is the mass of the flea

v is the take-off speed

g = 9.81 m/s^2 is the gravitational acceleration

`h=20 cm=0.20 m` is the maximum height reached by the flea

Solving the formula for v, we find

`v=√(2gh)=√(2(9.81 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 given by

`K_i = (1)/(2)mv^2`

where

`m=0.50 mg = 5\cdot 10^(-4)kg`

`v=1.98 m/s`

Substituting,

`K_i = (1)/(2)(5\cdot 10^(-4)kg)(1.98 m/s)^2=9.8\cdot 10^(-4)J`

While the kinetic energy per kilogram of mass is

`(K)/(m)=(9.8\cdot 10^(-4) J)/(5\cdot 10^(-4) kg)=1.96 J/kg`

(c) 200 m, 62.6 m/s

The flea jumps 0.20 m having a length of 2.0 mm = 0.002 m. The man has a length of 2.0 m, so in proportion he would jump at a maximum height of

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

And he should have a takeoff speed of

`v=√(2gh)=√(2(9.81 m/s^2)(200 m))=62.6 m/s`

(d) 5.88 J/kg

Most humans can jump to a maximum height of

h = 60 cm = 0.60 m

so the take off speed is

`v=√(2gh)=√(2(9.81 m/s^2)(0.60 m))=3.43 m/s`

So, its kinetic energy per kilogram of mass would be:

`(K)/(m)=((1)/(2)mv^2)/(m)=(v^2)/(2)=((3.43 m/s)^2)/(2)=5.88 J/kg`

(e) In the chemical and elastic potential energy store

The flea stores the energy that allows it to make the leap as chemical potential energy and elastic potential energy of its muscles, that are stretched and allows it to make the big jump.