A plane wave is described by the equation `y=3 cos ((x)/(4)-10 t-(pi)/(2))`. The maximum velocity of the particles of the medium due to this wave is

Options

A. 30
B. `(3 pi)/(2)`
C. `3//4`
D. 40

Answer

To find the maximum velocity of the particles of the medium due to the given wave described by the equation \( y = 3 \cos\left(\frac{x}{4} - 10t - \frac{\pi}{2}\right) \), we can follow these steps:

### Step-by-Step Solution

1. **Identify the Wave Equation**:
The wave is given by the equation:
\[ y = 3 \cos\left(\frac{x}{4} - 10t - \frac{\pi}{2}\right) \]

2. **Differentiate with Respect to Time**:
To find the velocity of the particles, we need to differentiate \( y \) with respect to time \( t \):
\[ v = \frac{dy}{dt} \]
Using the chain rule, we differentiate:
\[ v = \frac{d}{dt}[3 \cos\left(\frac{x}{4} - 10t - \frac{\pi}{2}\right)] \]
The derivative of \( \cos \) is \( -\sin \), so:
\[ v = -3 \sin\left(\frac{x}{4} - 10t - \frac{\pi}{2}\right) \cdot \frac{d}{dt}\left(\frac{x}{4} - 10t - \frac{\pi}{2}\right) \]
Since \( \frac{x}{4} \) is constant with respect to \( t \), its derivative is 0, and the derivative of \( -10t \) is \( -10 \):
\[ v = -3 \sin\left(\frac{x}{4} - 10t - \frac{\pi}{2}\right) \cdot (-10) = 30 \sin\left(\frac{x}{4} - 10t - \frac{\pi}{2}\right) \]

3. **Determine Maximum Velocity**:
The maximum value of \( \sin \) is 1. Therefore, the maximum velocity \( v_{\text{max}} \) is:
\[ v_{\text{max}} = 30 \cdot 1 = 30 \, \text{m/s} \]

4. **Conclusion**:
The maximum velocity of the particles of the medium due to this wave is:
\[ \boxed{30 \, \text{m/s}} \]