Question

The equation of a wave is $y=5\sin (\frac{t}{0.04}-\frac{x}{4}),$ where $x$ is in $cm$ and $t$ in seconds. The maximum velocity of the particle will be

• (A) 1 ms^-1
• (B) 2 ms^-1
• (C) 1.5 ms^-1
• (D) 1.25 ms^-1

Answer 1

The Correct Option is D

Explanation:
Equation of wave$$y=5\sin (\frac{t}{0.04}-\frac{x}{4})$$The standard equation of a wave in the given form is$$y=a\sin (\omega t-\frac{2\pi x}{\lambda })$$Comparing the given equation with the standard equation, we get$$a=5\text{ and }\omega =\frac{1}{0.04}=25$$Therefore, maximum velocity of particles of the medium,$$\begin{array}{rl}{v}_{max}& =a\omega \\ & =5\times 25\\ & =125{\mathrm{cms}}^{-1}=1.25{\mathrm{ms}}^{-1}\end{array}$$

Answer 2

A wave is a form of disturbance that travels through a medium due to the repeated periodic motion of the particles of the medium about their mean position.

The displacement equation of a wave traveling in +x direction is given by

y = a sin (kx - ωt)

Where

• a is the maximum amplitude
• k is the propagation constant or angular wave number
• ω is the angular frequency

Particle velocity

The velocity of the particle of media at any instant as a progressive wave travels through media is known as particle velocity.

Differentiating the displacement equation of progressive wave, we get

dy/dt = d/dt [a sin (kx - ωt)]

⇒ v = ωa cos (kx - ωt)

Above expression represents the particle velocity equation of a traveling wave. Where

• v = particle velocity at any instant t
• ω = angular frequency

The term ωa represents the maximum particle velocity. Therefore if v₀ is the maximum particle velocity, then the velocity equation becomes

v = v₀ cos (kx - ωt)

Where v₀ = ωa