A sine wave is travelling ina medium. The minium distance between the two particles, always having same speed is

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 always have the same speed, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Wave Equation**:
The displacement of a particle in a sine wave can be described by the equation:
\[ y(x, t) = A \sin(kx - \omega t) \]
where \( A \) is the amplitude, \( k \) is the wave number, and \( \omega \) is the angular frequency.

2. **Determine Particle Velocity**:
The velocity \( v \) of a particle in the wave can be derived from the displacement equation. The velocity is given by:
\[ v = \frac{\partial y}{\partial t} = \omega A \cos(kx - \omega t) \]
However, we can also express the speed of the particle in terms of its position \( x \):
\[ v = \omega \sqrt{A^2 - x^2} \]
This indicates that the speed depends on the position \( x \).

3. **Identify Points with Same Speed**:
To find two points where particles have the same speed, we can set the speeds of two particles at positions \( x_1 \) and \( x_2 \) equal to each other:
\[ \omega \sqrt{A^2 - x_1^2} = \omega \sqrt{A^2 - x_2^2} \]
This simplifies to:
\[ \sqrt{A^2 - x_1^2} = \sqrt{A^2 - x_2^2} \]
Squaring both sides gives:
\[ A^2 - x_1^2 = A^2 - x_2^2 \]
Thus:
\[ x_1^2 = x_2^2 \]

4. **Find the Minimum Distance**:
The solutions to \( x_1^2 = x_2^2 \) are \( x_1 = x_2 \) or \( x_1 = -x_2 \). The minimum distance between two particles having the same speed occurs when one particle is at \( x = A \) and the other at \( x = -A \). The distance between these two points is:
\[ d = |A - (-A)| = |A + A| = 2A \]

5. **Consider the Wave's Wavelength**:
The wavelength \( \lambda \) of the sine wave is related to the distance between repeating points in the wave. The minimum distance between two particles with the same speed occurs at half the wavelength:
\[ \text{Minimum distance} = \frac{\lambda}{2} \]

### Conclusion:
The minimum distance between two particles that always have the same speed in a sine wave is:
\[ \frac{\lambda}{2} \]