Question

Two small boats are 10m apart on a lake Each pops up and down with a period of 4.0 seconds due to wave motion on the surface of the water. When one boat is at its highest point, the other boat is at its lowest point. Both boats are always within a single cycle of the waves. The speed of the waves is:
$\begin{aligned} & (A)2.5m{{s}^{-1}} \\\ & (B)5m{{s}^{-1}} \\\ & (C)14m{{s}^{-1}} \\\ & (D)40m{{s}^{-1}} \\\ \end{aligned}$

Answer 1

It is given that both the boats are always within a cycle of waves. This means they are at two different positions of the same wave. When one is at its highest point, the other is at its lowest point. This means the distance between them is the distance between the crest and trough of the wave. Twice of this distance will give us the wavelength of the wave and we can divide it with the time period to get the speed of the wave.

Complete answer:
Let the distance between the boats be denoted by the term ‘d’. Since, the distance between these boats is also the distance between the successive crest and trough of a wave, therefore we can calculate the wavelength of the wave as follows:
$\Rightarrow$ Wavelength $=$ $2\times$ (The distance between successive crest and trough of a wave)
Hence, wavelength ($\lambda$) will be equal to:
$\Rightarrow \lambda =2d$
Here,
The value of ‘d’ is known to us and it is equal to 10 meters. Putting this value in the above equation, we get:
$\Rightarrow \lambda =2\times 10m$
$\Rightarrow \lambda =20m$ [Let this expression be equation number (1)]
Now, let the time period of wave be given by T. Then, it has been given as:
$\Rightarrow T=4s$ [Let this expression be equation number (1)]
Thus, the velocity of the wave (say V) can be calculated as:
$\Rightarrow V=\dfrac{\lambda }{T}$
Putting the values of known terms from equation number (1) and (2), we get:
$\begin{aligned} & \Rightarrow V=\dfrac{20}{4}m{{s}^{-1}} \\\ & \therefore V=5m{{s}^{-1}} \\\ \end{aligned}$
Hence, the velocity of waves forming on the surface of water is 5 meters per second.

Hence, option (B) is the correct option.

Note:
These are good practical examples of how concepts of physics are used to explain real life scenarios. This wave that forms on water is generally a circular wave and is commonly known as ripples. This can be observed when we throw a stone in still water. But due to loss of energy, overtime the wave disappears.