At t=0,a transverse wave pulse travelling in the positive x direction with a speed of `2 m//s` in a wire is described by the function `y=6//x^(2)` given that `x!=0`. Transverse velocity of a particle at x=2 m and t= 2 s is

Options

A. `3 m//s`
B. `-3 m//s`
C. `8 m//s`
D. `-8 m//s`

Explanation

To solve the problem step by step, we will follow the process outlined in the video transcript.

### Step 1: Understand the wave equation
The wave pulse is described by the function \( y = \frac{6}{x^2} \) at \( t = 0 \). The wave travels in the positive x-direction with a speed of \( v = 2 \, \text{m/s} \).

### Step 2: Write the general wave equation
The general form of a wave pulse traveling in the positive x-direction is given by:
\[ y(x, t) = \frac{A}{(x - vt)^2} \]
where \( A \) is a constant, \( v \) is the speed of the wave, \( x \) is the position, and \( t \) is the time.

Given that at \( t = 0 \), \( y = \frac{6}{x^2} \), we can identify \( A = 6 \) and write the wave equation as:
\[ y(x, t) = \frac{6}{(x - vt)^2} \]

### Step 3: Substitute the speed of the wave
Substituting \( v = 2 \, \text{m/s} \) into the equation gives:
\[ y(x, t) = \frac{6}{(x - 2t)^2} \]

### Step 4: Find the transverse velocity
The transverse velocity (particle velocity) is given by:
\[ v_p = \frac{\partial y}{\partial t} \]
To find this, we need to differentiate \( y \) with respect to \( t \).

### Step 5: Differentiate the wave equation
Using the chain rule, we differentiate \( y \):
\[ y = 6 (x - 2t)^{-2} \]
Differentiating with respect to \( t \):
\[ \frac{\partial y}{\partial t} = 6 \cdot (-2) \cdot (x - 2t)^{-3} \cdot (-2) = \frac{-24}{(x - 2t)^3} \]

### Step 6: Substitute \( x = 2 \, \text{m} \) and \( t = 2 \, \text{s} \)
Now we substitute \( x = 2 \) and \( t = 2 \) into the equation:
\[ v_p = \frac{-24}{(2 - 2 \cdot 2)^3} = \frac{-24}{(2 - 4)^3} = \frac{-24}{(-2)^3} = \frac{-24}{-8} = 3 \, \text{m/s} \]

### Final Answer
The transverse velocity of the particle at \( x = 2 \, \text{m} \) and \( t = 2 \, \text{s} \) is:
\[ v_p = -3 \, \text{m/s} \]