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

Explanation

To solve the problem, we need to find the speed of the wave traveling along the string. Here’s a step-by-step solution:

### Step 1: Understand the Formation of Nodes
When a wave reflects off a fixed end, it forms a standing wave. At the fixed end, there is a node (point of no displacement), and the next point where a node occurs is at a distance of 10 cm from the fixed end.

### Step 2: Determine the Distance Between Nodes
In a standing wave, the distance between two consecutive nodes is equal to half the wavelength (λ/2). Since we know that there is a node at 10 cm from the fixed end, the distance to the next node (the first antinode) is 10 cm.

### Step 3: Calculate the Wavelength
Since the distance between two nodes is λ/2, we can set up the equation:
\[ \frac{\lambda}{2} = 10 \, \text{cm} \]
To find the wavelength (λ), we multiply both sides by 2:
\[ \lambda = 2 \times 10 \, \text{cm} = 20 \, \text{cm} \]
Convert this to meters:
\[ \lambda = 20 \, \text{cm} = 0.20 \, \text{m} \]

### Step 4: Use the Wave Speed Formula
The speed (v) of a wave is given by the formula:
\[ v = f \cdot \lambda \]
where:
- \( f \) is the frequency of the wave,
- \( \lambda \) is the wavelength.

Given that the frequency \( f = 100 \, \text{Hz} \) and \( \lambda = 0.20 \, \text{m} \), we can substitute these values into the formula:
\[ v = 100 \, \text{Hz} \times 0.20 \, \text{m} = 20 \, \text{m/s} \]

### Step 5: Conclusion
The speed of the wave is \( 20 \, \text{m/s} \).

### Final Answer
The speed of the wave (incident and reflected) is **20 m/s**.

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