A progressive wave is represented by y = 5 `sin(100pit - 2pix)` where x and y are in m and t is in s. The maximum particle velocity is

Options

A. `100pims^(-1)`
B. `200pims^(-1)`
C. `300pims^(-1)`
D. `500pims^(-1)`

Explanation

To find the maximum particle velocity for the given progressive wave represented by the equation \( y = 5 \sin(100\pi t - 2\pi x) \), we will follow these steps:

### Step 1: Identify the wave equation
The wave equation is given as:
\[ y = 5 \sin(100\pi t - 2\pi x) \]
where \( y \) is the displacement, \( t \) is time, and \( x \) is the position.

### Step 2: Differentiate the wave equation with respect to time
To find the particle velocity \( v_p \), we need to differentiate the displacement \( y \) with respect to time \( t \):
\[ v_p = \frac{dy}{dt} \]
Using the chain rule, we differentiate:
\[ v_p = 5 \cdot \cos(100\pi t - 2\pi x) \cdot \frac{d}{dt}(100\pi t - 2\pi x) \]
Since \( x \) is constant with respect to \( t \), the derivative of \( -2\pi x \) with respect to \( t \) is 0. Thus, we have:
\[ \frac{d}{dt}(100\pi t - 2\pi x) = 100\pi \]
So, substituting this back, we get:
\[ v_p = 5 \cdot \cos(100\pi t - 2\pi x) \cdot 100\pi \]
This simplifies to:
\[ v_p = 500\pi \cos(100\pi t - 2\pi x) \]

### Step 3: Determine the maximum particle velocity
The maximum value of the cosine function is 1. Therefore, to find the maximum particle velocity \( v_{p_{max}} \):
\[ v_{p_{max}} = 500\pi \cdot 1 = 500\pi \]

### Step 4: Convert to numerical value
To express \( 500\pi \) in numerical terms, we can use the approximation \( \pi \approx 3.14 \):
\[ v_{p_{max}} \approx 500 \cdot 3.14 = 1570 \text{ m/s} \]

### Final Answer
Thus, the maximum particle velocity is:
\[ v_{p_{max}} = 500\pi \text{ m/s} \quad \text{(or approximately 1570 m/s)} \]

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