To find the velocity of the wave given by the equation \( y = 5 \, \text{mm} \, e^{\left(\frac{T}{5 \, \text{s}} - \frac{x}{5 \, \text{cm}}\right)^2} \), we can follow these steps:
### Step 1: Identify the coefficients
The general form of a wave equation can be expressed as:
\[ y = A e^{\left(\frac{T}{T_0} - \frac{x}{x_0}\right)^2} \]
where \( T_0 \) is the time constant and \( x_0 \) is the spatial constant.
From the given equation, we can identify:
- Coefficient of \( T \) (time) is \( \frac{1}{5 \, \text{s}} \)
- Coefficient of \( x \) (space) is \( \frac{1}{5 \, \text{cm}} \)
### Step 2: Write the formula for wave velocity
The velocity \( v \) of a wave can be calculated using the formula:
\[ v = -\frac{\text{coefficient of } T}{\text{coefficient of } x} \]
### Step 3: Substitute the coefficients into the formula
Now substituting the coefficients we identified:
- Coefficient of \( T \) = \( \frac{1}{5 \, \text{s}} \)
- Coefficient of \( x \) = \( \frac{1}{5 \, \text{cm}} \)
Thus,
\[ v = -\frac{\frac{1}{5 \, \text{s}}}{\frac{1}{5 \, \text{cm}}} \]
### Step 4: Simplify the expression
Now simplifying the expression:
\[ v = -\left(\frac{1}{5 \, \text{s}} \cdot \frac{5 \, \text{cm}}{1}\right) \]
\[ v = -1 \, \text{cm/s} \]
Since we are interested in the magnitude of the velocity, we take the absolute value:
\[ v = 1 \, \text{cm/s} \]
### Final Answer
The velocity of the wave is \( 1 \, \text{cm/s} \).
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