Question

A bug on the surface of a pond is observed to move up and down a total vertical distance of 7.0 cm, from the lowest to the highest point, as a wave passes. If the ripples decrease to 4.5 cm, by what factor does the bug’s maximum KE change?

Short answer

The bug’s maximum kinetic energy changes by a factor of \(0.41\).

Step-by-step solution

Understanding about the kinetic energy

Whenever a wave moves with a specific velocity, then the wave will have kinetic energy due to its amplitude.The relation between the kinetic energy and wave amplitude is a linear one.

Identification of given data

The given data can be listed below as,

• The total vertical displacement is\({A_1} = 7.0\;{\rm{cm}}\).
• The ripples decreases to the total vertical displacement of\({A_2} = 4.5\;{\rm{cm}}\).

Defining the expression of total mechanical energy of a wave

The expression of the total mechanical energy of the wave is given by,

\(E = \frac{1}{2}k{A^2}\)

Here,\(E\)is the total mechanical energy of the wave,\(A\)is the amplitude and\(k\)is wave number.

Determining the factor by which bug’s maximum kinetic energy change

From the above equation kinetic energy is directly proportional to the square of the amplitude. Hence,

\(\frac{{K{E_2}}}{{K{E_1}}} = {\left( {\frac{{{A_2}}}{{{A_1}}}} \right)^2}\)

Substitute all the known values in the above equation.

\(\begin{aligned}{c}\frac{{K{E_2}}}{{K{E_1}}} &= {\left( {\frac{{4.5\;{\rm{cm}}}}{{7\;{\rm{cm}}}}} \right)^2}\\ &\approx 0.41\end{aligned}\)

So, the bug’s maximum kinetic energy changes by a factor of \(0.41\).