In a standing wave experiment , a `1.2 - kg` horizontal rope is fixed in place at its two ends `( x = 0 and x = 2.0 m)` and made to oscillate up and down in the fundamental mode , at frequency of `5.0 Hz`. At `t = 0` , the point at `x = 1.0 m` has zero displacement and is moving upward in the positive direction of `y - axis` with a transverse velocity `3.14 m//s`.
Speed of the participating travelling wave on the rope is

Options

A. `6 m//s`
B. `15 m//s`
C. `20 m//s`
D. `24 m//s`

Explanation

To find the speed of the participating traveling wave on the rope, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Mass of the rope (not needed for this calculation): \( m = 1.2 \, \text{kg} \)
- Length of the rope: \( L = 2.0 \, \text{m} \)
- Frequency of oscillation: \( f = 5.0 \, \text{Hz} \)

2. **Determine the Wavelength**:
- In the fundamental mode of a standing wave on a rope fixed at both ends, the wavelength \( \lambda \) is related to the length of the rope by the formula:
\[ \lambda = 2L \]
- Substituting the length:
\[ \lambda = 2 \times 2.0 \, \text{m} = 4.0 \, \text{m} \]

3. **Calculate the Speed of the Wave**:
- The speed \( v \) of the wave can be calculated using the formula:
\[ v = f \times \lambda \]
- Substituting the values of frequency and wavelength:
\[ v = 5.0 \, \text{Hz} \times 4.0 \, \text{m} = 20.0 \, \text{m/s} \]

4. **Final Result**:
- The speed of the participating traveling wave on the rope is:
\[ \boxed{20.0 \, \text{m/s}} \]