A sine wave is travelling in a medium. A particular partile has zero displacement at a certain instant. The particle closest to it having zero displacement is at a distance

Options

A. `lamda/4`
B. `lamda/3`
C. `lamda/2`
D. `lamda`

Answer

To solve the problem of finding the minimum distance between two particles in a sine wave that have the same speed, we can follow these steps:

### Step 1: Understand the Wave Function
A sine wave can be described by the wave function:
\[ y(x, t) = A \sin(kx - \omega t) \]
where:
- \( y \) is the displacement of the wave,
- \( A \) is the amplitude,
- \( k \) is the wave number,
- \( x \) is the position,
- \( \omega \) is the angular frequency,
- \( t \) is the time.

### Step 2: Determine the Velocity of the Particles
The velocity \( v \) of a particle in the wave can be derived from the displacement function:
\[ v = \frac{\partial y}{\partial t} = \omega A \cos(kx - \omega t) \]
To find the speed, we need to consider the points where the displacement is zero.

### Step 3: Identify Points of Zero Displacement
The displacement \( y \) is zero when:
\[ A \sin(kx - \omega t) = 0 \]
This occurs at:
\[ kx - \omega t = n\pi \quad (n \in \mathbb{Z}) \]
Thus, the positions where displacement is zero are given by:
\[ x_n = \frac{n\pi}{k} + \frac{\omega t}{k} \]

### Step 4: Find Particles with Same Speed
The speed of particles is the same when they have the same value of \( \cos(kx - \omega t) \). The cosine function has the same value at:
\[ kx = n\pi \quad \text{and} \quad kx = (n + 1)\pi \]
This means that the particles at these positions will have the same speed.

### Step 5: Calculate the Minimum Distance
The minimum distance between two particles that have the same speed can be determined from the positions:
- The distance between \( x = \frac{n\pi}{k} \) and \( x = \frac{(n + 1)\pi}{k} \) is:
\[ \Delta x = \frac{(n + 1)\pi}{k} - \frac{n\pi}{k} = \frac{\pi}{k} \]

### Step 6: Relate to Wavelength
Since the wavelength \( \lambda \) is given by:
\[ \lambda = \frac{2\pi}{k} \]
The minimum distance between two particles with the same speed is:
\[ \Delta x = \frac{\lambda}{2} \]

### Conclusion
Thus, the minimum distance between two particles having the same speed in a sine wave is:
\[ \frac{\lambda}{2} \]