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Question

The equation of a transverse wave on a string is $y = A\sin(\omega t - kx)$.  The maximum transverse speed of a particle on the string is related to the wave speed $v$ by:

SUBHANI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: B
## CONCEPT FRAMEWORK
	**Primary Concept**: Wave Motion and Particle Velocity
	**Related Topics**: Simple Harmonic Motion, Wave Speed, Angular Frequency, Wave Number
	**Prerequisites**: Understanding of derivatives, trigonometric functions, wave parameters
## SOLUTION PATHWAY
**Given Information**:
	* Equation of transverse wave: $y = A\sin(\omega t - kx)$
	* Wave speed: $v$
	* Find the relation between maximum transverse speed of a particle and wave speed.
**Method & Steps**:
	1.	Find the transverse velocity of a particle on the string.
		Reasoning: The transverse velocity is the time derivative of the transverse displacement.
		Formula/Rule: $v_y = \frac{\partial y}{\partial t}$
	2.	Calculate the derivative.
		Working: $v_y = \frac{\partial}{\partial t} [A\sin(\omega t - kx)] = A\omega \cos(\omega t - kx)$
		Key Point: The maximum transverse velocity occurs when $\cos(\omega t - kx) = 1$.
	3.	Find the maximum transverse speed.
		Result: $v_{y,max} = A\omega$
	4.	Relate the maximum transverse speed to the wave speed.
		Working: The wave speed is given by $v = \frac{\omega}{k}$. Therefore, $\omega = vk$.  Substituting this into $v_{y,max}$, we get $v_{y,max} = A(vk) = (Ak)v$. 
		Verification: The maximum transverse speed is proportional to the wave speed.
 ## OPTION ANALYSIS
[A] :x: The maximum transverse speed is not equal to the wave speed. It is proportional to the wave speed.
[B] ✓: The maximum transverse speed, $A\omega$, is proportional to the wave speed $v$ since $v_{y,max}=Ak v$ and $A$ and $k$ are constants for a given wave.
[C] :x: The maximum transverse speed is not inversely proportional to the wave speed. It is proportional to it.
[D] :x: The maximum transverse speed is dependent on the wave speed; it is proportional to it.
 ## LEARNING AIDS
	**Common Mistakes**:
		1.[Error 1]: Confusing particle velocity with wave velocity. Particle velocity is the velocity of the individual particles oscillating up and down, while wave velocity is the speed of the disturbance moving along the string.
		2.[Error 2]: Forgetting to take the derivative with respect to time when finding particle velocity.
	**Quick Tips**:
		1.[Helpful shortcut]: Remember that maximum values of sine and cosine are 1. Use this to directly determine the maximum transverse velocity.
		2.[Memory aid]: The maximum transverse speed is proportional to the wave speed, and the proportionality constant is the product of the amplitude and the wave number.

SUBHANI · Answer

equal to

SUBHANI · Answer

The correct Option is "proportional to"

SUBHANI · Answer

inversely proportional to

SUBHANI · Answer

independent of

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