A motion is described by `y = 3e^(x).e^(-3t)` where y,x are in metre and t is in second :

Options

A. This represents equation of progressive wavepropagating along - X direction with `3ms^(-1)`
B. This represents equation of progressive wave propagating along + X direction with `3ms^(-1)`
C. This does not represent a progressive wave equation
D. Data is insufficient to arrive at any conclusion ofthis short

Explanation

To determine if the motion described by the equation \( y = 3e^{x} e^{-3t} \) represents a progressive wave propagating along the x-direction with a speed of 3 m/s, we will follow these steps:

### Step 1: Identify the form of the equation
The given equation is:
\[ y = 3e^{x} e^{-3t} \]
This can be rewritten as:
\[ y = 3 e^{x - 3t} \]
This form suggests that it could represent a wave, as it combines spatial and temporal components.

### Step 2: Calculate the first and second derivatives with respect to time
We first need to find the first derivative of \( y \) with respect to \( t \):
\[ \frac{\partial y}{\partial t} = 3 e^{x} \frac{\partial}{\partial t}(e^{-3t}) = 3 e^{x} (-3 e^{-3t}) = -9 e^{x} e^{-3t} \]

Next, we calculate the second derivative with respect to time:
\[ \frac{\partial^2 y}{\partial t^2} = \frac{\partial}{\partial t}(-9 e^{x} e^{-3t}) = -9 e^{x} \frac{\partial}{\partial t}(e^{-3t}) = -9 e^{x} (-3 e^{-3t}) = 27 e^{x} e^{-3t} \]

### Step 3: Calculate the first and second derivatives with respect to space
Now we find the first derivative of \( y \) with respect to \( x \):
\[ \frac{\partial y}{\partial x} = 3 e^{x} e^{-3t} \quad \text{(since \( e^{-3t} \) is constant with respect to \( x \))} \]

Next, we calculate the second derivative with respect to space:
\[ \frac{\partial^2 y}{\partial x^2} = 3 e^{x} e^{-3t} \quad \text{(same reasoning as above)} \]

### Step 4: Check if the wave equation is satisfied
The wave equation in its second differential form is given by:
\[ \frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2} \]
Substituting the derivatives we calculated:
\[ 27 e^{x} e^{-3t} = v^2 (3 e^{x} e^{-3t}) \]
We can simplify this:
\[ 27 = v^2 \cdot 3 \]
Dividing both sides by 3 gives:
\[ v^2 = 9 \]
Taking the square root, we find:
\[ v = 3 \text{ m/s} \]

### Step 5: Determine the direction of propagation
The general form of a wave traveling in the positive x-direction is \( e^{(kx - \omega t)} \). Here, we have \( y = 3 e^{(x - 3t)} \), which indicates that the wave is indeed propagating in the positive x-direction.

### Conclusion
Thus, the motion described by the equation \( y = 3e^{x} e^{-3t} \) does represent a progressive wave propagating along the x-direction with a speed of 3 m/s.