The equation of a wave pulse moving with a speed 1 m/sec at time t = 0 is given as `y=f(x)=1/(1+x^(2))`. Its equation at time t = 1 second can be given as

Options

A. `y=1/([1+(1+x)]^(2))`
B. `y=1/(1+(1-x)^(2))`
C. `y=1/(1+(x^(2)+1))`
D. `y=1/(1+(1/(1+x^(2))))`

Answer

To find the equation of the wave pulse at time \( t = 1 \) second, we start with the initial wave pulse equation given at \( t = 0 \):

\[ y = f(x) = \frac{1}{1 + x^2} \]

The wave pulse is moving with a speed of \( v = 1 \, \text{m/s} \). The general form of the wave function at a later time \( t \) can be expressed as:

\[ f(x - vt) \]

Where:
- \( v \) is the speed of the wave,
- \( t \) is the time,
- \( x \) is the position.

### Step 1: Substitute the values into the wave equation

At \( t = 1 \) second, we substitute \( v = 1 \, \text{m/s} \) and \( t = 1 \) into the equation:

\[ f(x - vt) = f(x - 1 \cdot 1) = f(x - 1) \]

### Step 2: Replace \( x \) in the original function

Now, we need to replace \( x \) in the original function \( f(x) \) with \( x - 1 \):

\[ f(x - 1) = \frac{1}{1 + (x - 1)^2} \]

### Step 3: Simplify the expression

Now we simplify the expression:

\[ f(x - 1) = \frac{1}{1 + (x^2 - 2x + 1)} = \frac{1}{x^2 - 2x + 2} \]

Thus, the equation of the wave pulse at \( t = 1 \) second is:

\[ y = \frac{1}{x^2 - 2x + 2} \]

### Final Answer:
The equation of the wave pulse at time \( t = 1 \) second is:

\[ y = \frac{1}{x^2 - 2x + 2} \]

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