A motion is described by `Y = 4e^(x) (e ^- (5t))` , Where y,x are in meters and t is in second .

Options

A. This represents progressive wave propagating along - x direction with `5 ms ^(-1)`
B. This represents progressive wave propagating along + x direction with `5 ms ^(-1)`
C. This does not represent progressive wave.
D. This represents standing wave.

Answer

To solve the problem given by the equation \( Y = 4e^{x} e^{-5t} \), we need to analyze the motion described by this equation in terms of wave propagation.

### Step-by-Step Solution:

1. **Identify the wave function format**:
The general form of a progressive wave can be expressed as \( Y = A e^{(ax - bt)} \), where:
- \( A \) is the amplitude,
- \( a \) is the coefficient of \( x \),
- \( b \) is the coefficient of \( t \).

In our case, we can rewrite the given equation:
\[ Y = 4 e^{(x - 5t)} \]
Here, we can identify:
- \( a = 1 \) (coefficient of \( x \)),
- \( b = 5 \) (coefficient of \( t \)).

2. **Determine the direction of wave propagation**:
- If \( a \) and \( b \) have the same sign, the wave travels in the negative x-direction.
- If \( a \) and \( b \) have different signs, the wave travels in the positive x-direction.

In our case:
- \( a = 1 \) (positive),
- \( b = 5 \) (positive).

Since both \( a \) and \( b \) are positive, the wave travels in the negative x-direction.

3. **Calculate the velocity of the wave**:
The velocity \( v \) of the wave can be calculated using the formula:
\[ v = -\frac{b}{a} \]
Substituting the values:
\[ v = -\frac{5}{1} = -5 \text{ m/s} \]
The negative sign indicates the direction of the wave, which we already established is in the negative x-direction.

### Summary:
- The wave travels in the **negative x-direction**.
- The velocity of the wave is **5 m/s**.