A wave frequency `100 Hz` travels along a string towards its fixed end . When this wave travels back after reflection , a node is formed at a distance of `10 cm `from the fixed end . The speed of the wave (incident and reflected) is

Options

A. 5m/s
B. 10 m/s
C. 20 m/s
D. 40 m/s

Answer

To find the speed of the wave traveling along the string, we can follow these steps:

### Step 1: Understand the Problem
We know that a wave with a frequency of 100 Hz travels along a string towards a fixed end. After reflection, a node is formed at a distance of 10 cm from the fixed end.

### Step 2: Identify the Distance of the Node
The distance from the fixed end to the node is given as 10 cm. We need to convert this distance into meters for consistency in units:
\[ 10 \, \text{cm} = 0.1 \, \text{m} \]

### Step 3: Determine the Wavelength
In a standing wave formed on a string with one end fixed, the distance to the first node is one-quarter of the wavelength (λ). Therefore, if the distance to the first node is 0.1 m, we can express the wavelength as:
\[ \text{Distance to node} = \frac{\lambda}{4} \]
Thus,
\[ 0.1 \, \text{m} = \frac{\lambda}{4} \]
To find the wavelength (λ), we multiply both sides by 4:
\[ \lambda = 4 \times 0.1 \, \text{m} = 0.4 \, \text{m} \]

### Step 4: Use the Wave Speed Formula
The speed of a wave (v) can be calculated using the formula:
\[ v = f \times \lambda \]
Where:
- \( f \) is the frequency of the wave (100 Hz)
- \( \lambda \) is the wavelength (0.4 m)

### Step 5: Calculate the Speed
Substituting the values into the formula:
\[ v = 100 \, \text{Hz} \times 0.4 \, \text{m} \]
Calculating this gives:
\[ v = 40 \, \text{m/s} \]

### Final Answer
The speed of the wave (both incident and reflected) is:
\[ \boxed{40 \, \text{m/s}} \]

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