Answer to Question #169509 in Mechanics | Relativity for Gumanit Eva D.

Question

A speedboat sends out a wave that rocks a small boat where a senior high school student is standing. She notices that the height of the wave as measured from trough to peak is 0.60 m and that the wave rocks the small boat with a period of 2.0 s. The distance between crests is 1.9m. What are the amplitude,wavelength,the frequency,wave speed, wave number angular frequency ,wave function and the maximum vertical speed of the rowboat

Answer 1

(a) The amplitude of the wave is the distance from the rest position to crest of the wave (or from the rest to the trough). Since, we know that the height of the wave measured from trough to peak (or crest) is 0.60 m, the amplitude of the wave equals:

$$A=\dfrac{0.6\ m}{2}=0.3\ m.$$

(b) The wavelength of the wave is the distance between two crests:

$$\lambda=1.9\ m.$$

(c) The frequency can be found as follows:

$$f=\dfrac{1}{T}=\dfrac{1}{2\ s}=0.5\ Hz.$$

(d) Wave speed can be found from the wave speed equation:

$$v=f\lambda=0.5\ Hz\cdot1.9\ m=0.95\ \dfrac{m}{s}.$$

(e) Wave number can be found as follows:

$$k=\dfrac{2\pi}{\lambda}=\dfrac{2\pi}{1.9\ m}=3.3\ m^{-1}.$$

(f) Angular frequency can be found as follows:

$$\omega=2\pi f=2\pi\cdot0.5\ Hz=3.14\ \dfrac{rad}{s}.$$

(g) A general form of wave function can be written as follows:

$$y(x,t)=Asin(kx-\omega t).$$

Substituting amplitude, wave number and angular frequency, we get:

$$y(x,t)=3sin(3.3x-3.14t).$$

(h) Let's take derivative of position with respect to time and find the speed of the boat:

$$v=\dfrac{dy}{dt}=-3\ m\cdot3.14\ \dfrac{rad}{s} \cdot cos(kx-\omega t)=-9.42\ \dfrac{m}{s}\cdot cos(kx-\omega t).$$

The maximum  vertical speed of the rowboat occurs when $cos(kx-\omega t)=1$. Therefore, the magnitude of the vertical speed of the rowboat equals:

$$v_{max}=9.42\ \dfrac{m}{s}.$$