The displacement of a particle of a string carrying a travelling wave is given by `y=(4cm)sin2pi(0.5x-100t)`, where x is in cm and t is in seconds. The speed of the wave is

Options

A. 50 cm/s
B. 100 cm/s
C. 200 cm/s
D. 250 cm/s

Explanation

To find the speed of the wave described by the equation \( y = (4 \, \text{cm}) \sin(2\pi(0.5x - 100t)) \), we can follow these steps:

### Step 1: Identify the wave equation components
The given wave equation is in the form:
\[ y = A \sin(kx - \omega t) \]
where:
- \( A \) is the amplitude,
- \( k \) is the wave number (propagation constant),
- \( \omega \) is the angular frequency.

From the equation \( y = (4 \, \text{cm}) \sin(2\pi(0.5x - 100t)) \), we can identify:
- \( A = 4 \, \text{cm} \)
- \( k = 2\pi \times 0.5 \)
- \( \omega = 2\pi \times 100 \)

### Step 2: Calculate the wave number \( k \)
Using the identified value of \( k \):
\[ k = 2\pi \times 0.5 = \pi \, \text{cm}^{-1} \]

### Step 3: Calculate the angular frequency \( \omega \)
Using the identified value of \( \omega \):
\[ \omega = 2\pi \times 100 = 200\pi \, \text{s}^{-1} \]

### Step 4: Use the relationship between wave speed, wave number, and angular frequency
The speed of the wave \( v \) can be calculated using the formula:
\[ v = \frac{\omega}{k} \]

### Step 5: Substitute the values of \( \omega \) and \( k \)
Substituting the values we calculated:
\[ v = \frac{200\pi}{\pi} = 200 \, \text{cm/s} \]

### Final Answer
The speed of the wave is:
\[ v = 200 \, \text{cm/s} \]

---