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?

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.

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\({A_2} = 4.5\;{\rm{cm}}\).

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.

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 \(0.41\).