A progressive wave having amplitude `5 m` and wavelength `3m`. If the maximum average velocity of particle in half time period is `5 m//s` and waves is moving in the positive x-direction then find which may be the correct equation (s) of the wave ? [wave `x` in meter].

Options

A. `5 sin ((2pi)/(5)t-(2pi)/(3)x)`
B. `4 sin((pit)/(2)-(2pi)/(3))+3 cos ((pi t)/(2)-(2pi)/(3)x)`
C. `5 sin ((pi t)/(2)-(2pi)/(3)x)`
D. `3 cos((2pi)/(5)t-(2pi)/(3)x)-4 sin ((2pi)/(5)t-(2pi)/(3)x)`

Explanation

To solve the problem, we need to find the equation of a progressive wave given its amplitude, wavelength, and the maximum average velocity of the particles.

### Step 1: Identify the given parameters
- Amplitude (A) = 5 m
- Wavelength (λ) = 3 m
- Maximum average velocity of particle in half time period (V_avg) = 5 m/s

### Step 2: Relate the average velocity to amplitude and time period
The average velocity of a particle in half a time period can be expressed as:
\[ V_{\text{avg}} = \frac{2A}{T/2} = \frac{4A}{T} \]
Substituting the known values:
\[ 5 = \frac{4 \times 5}{T} \]
This simplifies to:
\[ 5 = \frac{20}{T} \]

### Step 3: Solve for the time period (T)
Rearranging the equation gives:
\[ T = \frac{20}{5} = 4 \text{ seconds} \]

### Step 4: Calculate the angular frequency (ω)
The angular frequency (ω) is related to the time period (T) by the formula:
\[ \omega = \frac{2\pi}{T} \]
Substituting the value of T:
\[ \omega = \frac{2\pi}{4} = \frac{\pi}{2} \text{ rad/s} \]

### Step 5: Calculate the wave number (k)
The wave number (k) is given by:
\[ k = \frac{2\pi}{\lambda} \]
Substituting the value of λ:
\[ k = \frac{2\pi}{3} \text{ rad/m} \]

### Step 6: Write the equation of the wave
The general equation of a progressive wave moving in the positive x-direction is given by:
\[ y = A \sin(\omega t - kx) \]
Substituting the values of A, ω, and k:
\[ y = 5 \sin\left(\frac{\pi}{2} t - \frac{2\pi}{3} x\right) \]

### Conclusion
Thus, the equation of the wave is:
\[ y = 5 \sin\left(\frac{\pi}{2} t - \frac{2\pi}{3} x\right) \]