The equation of a transverse wave is`y=asin2pi[t-(x//5)]` , then the ratio of maximum particle velocity and wave velocity is

Options

A. `(2pia)/(sqrt(5))`
B. `(2pia)/(5)`
C. `(a)/(5)`
D. `2piasqrt(5)`

Explanation

To solve the problem, we need to find the ratio of the maximum particle velocity to the wave velocity from the given wave equation \( y = a \sin(2\pi [t - \frac{x}{5}]) \).

### Step-by-Step Solution:

1. **Identify the wave equation**: The given equation is \( y = a \sin(2\pi [t - \frac{x}{5}]) \). We can rewrite this in the standard form of a wave equation, which is \( y = a \sin(\omega t - kx) \).

2. **Extract parameters**: From the equation, we can identify:
- Angular frequency \( \omega = 2\pi \)
- Wave number \( k = \frac{2\pi}{5} \)

3. **Calculate maximum particle velocity**: The maximum particle velocity \( V_{\text{max}} \) is given by the formula:
\[ V_{\text{max}} = a \omega \]
Substituting the value of \( \omega \):
\[ V_{\text{max}} = a \cdot 2\pi \]

4. **Calculate wave velocity**: The wave velocity \( v \) is calculated using the formula:
\[ v = \frac{\omega}{k} \]
Substituting the values of \( \omega \) and \( k \):
\[ v = \frac{2\pi}{\frac{2\pi}{5}} = 5 \]

5. **Find the ratio of maximum particle velocity to wave velocity**:
\[ \text{Ratio} = \frac{V_{\text{max}}}{v} = \frac{a \cdot 2\pi}{5} \]

6. **Final expression**: The ratio of maximum particle velocity to wave velocity is:
\[ \text{Ratio} = \frac{2\pi a}{5} \]

### Conclusion:
The ratio of maximum particle velocity to wave velocity is \( \frac{2\pi a}{5} \).