Question

A boat at an anchor is rocked by waves whose crests are 100 m apart. The wave velocity of the moving crest is 25m/s. the boat bounces up at every:
A. 2500s
B. 75s
C. 4s
D. 0.25s

Answer 1

The distance between the two consecutive crests in a wave motion is called the wavelength. The boat bounces up, i.e., it travels from the crest to the consecutive crest along with wave motion. Wavelength =distance between two consecutive crests. Frequency and wavelength are inversely proportional to each other. The wave with the greatest frequency has the shortest wavelength.

Formula used: Time period (T) is used to describe a wave. It is defined as the time taken to complete oscillation. Because frequency determines the number of times a wave oscillates and mathematically it can be expressed as:
$f = \dfrac{1}{T}$
Wave velocity is given by the product of frequency and wavelength. Mathematically:
$v = \lambda f$

Complete Answer:
Given:
Wavelength of the wave $= \lambda = 100m$
Velocity of the wave$= v = 25m/s$
We know that,
Wave velocity is given by,
$v = f \times \lambda$
Where,
$f =$Frequency of waves.
$\Rightarrow f = \dfrac{v}{\lambda }$
Substituting the values of velocity and wavelength in the above equation we get:
$\Rightarrow f = \dfrac{{25}}{{100}}Hz$
We know that the time period is defined as the reciprocal of the frequency of the given wave.
Then the time period is given by:
$T = \dfrac{{100}}{{25}} = 4s$
Hence, the boat will bounce up at every 4s.
Therefore, the correct option is C.

Note: Time period (T) can be calculated directly by using the formula, i.e., wave velocity is directly proportional to wavelength and inversely proportional to time period:
$v = \dfrac{\lambda }{T}$
$\Rightarrow T = \dfrac{\lambda }{v}$
$T = \dfrac{{100}}{{25}} = 4s$
Both methods can be used in order to calculate the time period.