A simple harmonic wave is represented as `y = "5 sin" (pi)/(2) (100 t - 2x)` x, y are in metres, t in seconds.
The ratio of maximum particle velocity to wave velocity will be

Options

A. `2pi`
B. `3 pi`
C. `4 pi`
D. `5 pi`

Explanation

To find the ratio of maximum particle velocity to wave velocity for the given wave equation \( y = 5 \sin\left(\frac{\pi}{2}(100t - 2x)\right) \), we can follow these steps:

### Step 1: Identify the wave equation parameters
The general form of a wave equation is:
\[ y = A \sin(\omega t - kx) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number.

From the given equation \( y = 5 \sin\left(\frac{\pi}{2}(100t - 2x)\right) \), we can identify:
- Amplitude \( A = 5 \)
- Coefficient of \( t \) (which gives us \( \omega \)) is \( \frac{\pi}{2} \times 100 = 50\pi \)
- Coefficient of \( x \) (which gives us \( k \)) is \( \frac{\pi}{2} \times 2 = \pi \)

### Step 2: Calculate wave velocity (\( v_w \))
The wave velocity \( v_w \) can be calculated using the formula:
\[ v_w = \frac{\omega}{k} \]
Substituting the values we found:
\[ v_w = \frac{50\pi}{\pi} = 50 \, \text{m/s} \]

### Step 3: Calculate maximum particle velocity (\( v_{pm} \))
The maximum particle velocity \( v_{pm} \) can be calculated using the formula:
\[ v_{pm} = A \omega \]
Substituting the values of \( A \) and \( \omega \):
\[ v_{pm} = 5 \times 50\pi = 250\pi \, \text{m/s} \]

### Step 4: Calculate the ratio of maximum particle velocity to wave velocity
Now, we can find the ratio:
\[ \text{Ratio} = \frac{v_{pm}}{v_w} = \frac{250\pi}{50} = 5\pi \]

### Final Answer
The ratio of maximum particle velocity to wave velocity is:
\[ \boxed{5\pi} \]

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