A parabolic pulse given by equation `y ("in cm") = 0.3 - 0.1(x-5t)^(2) (y ge 0)` travelling in a uniform string. The pulse passes through a boundary beyond which its velocity becomes `2.5 m//s`. What will be the amplitude of pulse in this medium after transmission ?

Answer

To solve the problem, we need to determine the amplitude of the parabolic pulse after it passes through a boundary where its velocity changes. Let's break down the solution step by step.

### Step 1: Identify the given parameters
The equation of the pulse is given as:
\[ y = 0.3 - 0.1(x - 5t)^2 \]
From this equation, we can identify:
- The initial amplitude \( A_1 = 0.3 \, \text{cm} \)
- The initial velocity \( v_1 = 5 \, \text{m/s} \) (coefficient of \( t \))

The new velocity after the boundary is given as:
\[ v_2 = 2.5 \, \text{m/s} \]

### Step 2: Use the amplitude transmission formula
The amplitude of a wave changes when it passes through a boundary, and the relationship between the amplitudes and velocities can be expressed as:
\[ A_2 = A_1 \cdot \frac{v_1}{v_1 + v_2} \]

### Step 3: Substitute the values into the formula
Now, substituting the known values into the formula:
- \( A_1 = 0.3 \, \text{cm} \)
- \( v_1 = 5 \, \text{m/s} \)
- \( v_2 = 2.5 \, \text{m/s} \)

We can calculate:
\[ A_2 = 0.3 \cdot \frac{5}{5 + 2.5} \]

### Step 4: Simplify the expression
First, calculate \( 5 + 2.5 \):
\[ 5 + 2.5 = 7.5 \]

Now substitute back into the equation:
\[ A_2 = 0.3 \cdot \frac{5}{7.5} \]

### Step 5: Calculate the final amplitude
Now, simplify \( \frac{5}{7.5} \):
\[ \frac{5}{7.5} = \frac{5 \times 2}{7.5 \times 2} = \frac{10}{15} = \frac{2}{3} \]

Now, substitute this back into the equation for \( A_2 \):
\[ A_2 = 0.3 \cdot \frac{2}{3} = 0.3 \cdot 0.6667 \approx 0.2 \, \text{cm} \]

### Final Answer
Thus, the amplitude of the pulse in the new medium after transmission is:
\[ A_2 = 0.2 \, \text{cm} \]

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