A person standing between two parallel hills fires a gun. He hears the first echo after `(3) /(2)` s, and a second echo after `(5) /(2)` s. If speed of sound is `332 m//s` , Calculate the distance between the hills. When will he hear the third echo?

Explanation

To solve the problem step by step, we will first analyze the situation and then calculate the required distances and the time for the third echo.

### Step 1: Understanding the Echoes
When the person fires the gun, the sound travels to the first hill and reflects back, which is heard as the first echo. The time taken for the first echo is given as \( \frac{3}{2} \) seconds. The second echo, which is the sound traveling to the second hill and back, takes \( \frac{5}{2} \) seconds.

### Step 2: Calculating the Distance to the First Hill (D1)
The distance to the first hill can be calculated using the formula:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
Since the sound travels to the hill and back, the total distance covered is \( 2D_1 \). Therefore:
\[ 2D_1 = \text{Speed of Sound} \times \text{Time for First Echo} \]
Substituting the values:
\[ 2D_1 = 332 \, \text{m/s} \times \frac{3}{2} \, \text{s} \]
Calculating \( 2D_1 \):
\[ 2D_1 = 332 \times 1.5 = 498 \, \text{m} \]
Now, solving for \( D_1 \):
\[ D_1 = \frac{498}{2} = 249 \, \text{m} \]

### Step 3: Calculating the Distance to the Second Hill (D2)
Similarly, for the second echo:
\[ 2D_2 = \text{Speed of Sound} \times \text{Time for Second Echo} \]
Substituting the values:
\[ 2D_2 = 332 \, \text{m/s} \times \frac{5}{2} \, \text{s} \]
Calculating \( 2D_2 \):
\[ 2D_2 = 332 \times 2.5 = 830 \, \text{m} \]
Now, solving for \( D_2 \):
\[ D_2 = \frac{830}{2} = 415 \, \text{m} \]

### Step 4: Calculating the Total Distance Between the Hills
The total distance between the two hills is:
\[ \text{Total Distance} = D_1 + D_2 \]
Substituting the values:
\[ \text{Total Distance} = 249 \, \text{m} + 415 \, \text{m} = 664 \, \text{m} \]

### Step 5: Calculating the Time for the Third Echo
The third echo will occur after the sound travels to the first hill and back, then to the second hill and back again. The time for the third echo can be calculated as:
\[ \text{Time for Third Echo} = \text{Time for First Echo} + \text{Time for Second Echo} \]
Substituting the values:
\[ \text{Time for Third Echo} = \frac{3}{2} \, \text{s} + \frac{5}{2} \, \text{s} = \frac{8}{2} \, \text{s} = 4 \, \text{s} \]

### Final Answers
- The distance between the hills is \( 664 \, \text{m} \).
- The time when he will hear the third echo is \( 4 \, \text{s} \).