The equation of a simple harmonic progressive wave is given by
`y = 5 cos pi [200t - (x)/(150)]`
where x and y are in cm and 't' is in seocnd. The the velocity of the wave is

Options

A. 2 m/s
B. 200 m/s
C. 300 m/s
D. 150 m/s

Explanation

To find the velocity of the wave given by the equation \( y = 5 \cos \left( \pi \left( 200t - \frac{x}{150} \right) \right) \), we can follow these steps:

### Step 1: Identify the wave equation format
The standard form of a simple harmonic progressive wave is:
\[ y = A \cos(\omega t - kx) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number.

### Step 2: Rewrite the given equation
The given wave equation is:
\[ y = 5 \cos \left( \pi \left( 200t - \frac{x}{150} \right) \right) \]
We can expand this to:
\[ y = 5 \cos \left( 200\pi t - \frac{\pi}{150} x \right) \]

### Step 3: Identify parameters
From the equation, we can identify:
- Amplitude \( A = 5 \) cm,
- Angular frequency \( \omega = 200\pi \) rad/s,
- Wave number \( k = \frac{\pi}{150} \) rad/cm.

### Step 4: Calculate frequency
The relationship between angular frequency and frequency \( f \) is given by:
\[ \omega = 2\pi f \]
Substituting the value of \( \omega \):
\[ 200\pi = 2\pi f \]
Dividing both sides by \( 2\pi \):
\[ f = \frac{200\pi}{2\pi} = 100 \text{ Hz} \]

### Step 5: Calculate wavelength
The wave number \( k \) is related to the wavelength \( \lambda \) by:
\[ k = \frac{2\pi}{\lambda} \]
Substituting the value of \( k \):
\[ \frac{\pi}{150} = \frac{2\pi}{\lambda} \]
Cross-multiplying gives:
\[ \lambda = 2 \times 150 = 300 \text{ cm} \]

### Step 6: Calculate wave velocity
The velocity \( v \) of the wave can be calculated using the formula:
\[ v = f \lambda \]
Substituting the values of \( f \) and \( \lambda \):
\[ v = 100 \text{ Hz} \times 300 \text{ cm} = 30000 \text{ cm/s} \]
Converting to meters per second (since \( 1 \text{ m} = 100 \text{ cm} \)):
\[ v = \frac{30000 \text{ cm/s}}{100} = 300 \text{ m/s} \]

### Final Answer
The velocity of the wave is \( 300 \text{ m/s} \).

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