Question

Fleas have remarkable jumping ability. a 0.50 mg flea, jumping straight up, would reach a height of 40 cm if there were no air resistance. in reality, air resistance limits the height to 20 cm.
a. what is the flea's kinetic energy as it leaves the ground?
b. at its highest point, what fraction of the initial kinetic energy has been converted to potential energy?

Answer 1

The theorem of conservation of mechanical energy is used

a) The initial kinetic energy is, K = 1.33 10⁻⁶ J

b) The fraction of kinetic energy converted to potential energy, U/K = 1.96

Calculation:

The total mechanical energy which is defined as the sum of kinetic energy plus potential energies, is given as

                                            E = K + U

To find the speed of the flea jump when there is no friction,

Initial point

E₁ = K = ½ m v²

Final point.

At the highest height h = 30 cm = 0.30 m because there is no friction

E₂ = U = mgh

Energy is conserved

E₁ = E₂

½ mv² = mgh

v = $\sqrt{2gh}$

v = $\sqrt{2(10)(0.4)}$

v = $\sqrt{4}$ m/s

v = 2 m/s

A) Now we have to find the initial kinetic energy

K = ½ m v²

K = ½ (0.5 × 10⁻⁶) (2)²

K = 1 × 10⁻⁶ J

B) Now we have to find the fraction of energy lost if we take into account the friction of the air

The initial kinetic energy

K = ½ m v²

Now we have to look for the potential energy until h’= 20 cm = 0.20 m

U = mgh'

The fraction of energy lost,

U/K = mgh'/½ mv²

      = 2gh'/v²

      = 2 × 9.8 × 0.2/ 2

      = 1.96

Therefore,

a) The initial kinetic energy is, K = 1 × 10⁻⁶ J

b) The fraction of kinetic energy converted to potential energy, U/K = 1.96

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