A soldier fires a bullet towards a fort-wall and hears the first echo after 2s. He moves a distance of 85m towards the wall, fires a bullet and hears the echo after 1.5 s. The velocity of sound is

Options

A. 330 m/s
B. 340 m/s
C. 345 m/s
D. none

Explanation

To solve the problem step by step, we need to analyze the situation described in the question.

### Step 1: Understanding the First Echo
When the soldier fires the bullet towards the wall, the sound travels to the wall and back. The total time taken for this journey is 2 seconds.

Let the distance from the soldier to the wall be \( x \) meters. The sound travels a total distance of \( 2x \) meters (to the wall and back).

Using the formula for speed:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
we can express the speed of sound \( v \) as:
\[ v = \frac{2x}{2} = x \quad \text{(1)} \]

### Step 2: Understanding the Second Echo
After moving 85 meters closer to the wall, the soldier fires another bullet and hears the echo after 1.5 seconds. Now, the distance from the soldier to the wall is \( x - 85 \) meters.

The total distance the sound travels in this case is \( 2(x - 85) \) meters (to the wall and back). The time taken for this journey is 1.5 seconds.

Using the speed formula again, we can express the speed of sound as:
\[ v = \frac{2(x - 85)}{1.5} \quad \text{(2)} \]

### Step 3: Setting Up the Equations
From equation (1), we have:
\[ v = x \]

From equation (2), we can rewrite it as:
\[ v = \frac{4(x - 85)}{3} \]

### Step 4: Equating the Two Expressions for Velocity
Since both expressions equal \( v \), we can set them equal to each other:
\[ x = \frac{4(x - 85)}{3} \]

### Step 5: Solving for \( x \)
To eliminate the fraction, multiply both sides by 3:
\[ 3x = 4(x - 85) \]

Expanding the right side:
\[ 3x = 4x - 340 \]

Rearranging gives:
\[ 340 = 4x - 3x \]
\[ 340 = x \]

### Step 6: Finding the Velocity of Sound
Now that we have \( x = 340 \) meters, we can find the velocity of sound using equation (1):
\[ v = x = 340 \text{ m/s} \]

### Final Answer
The velocity of sound is \( 340 \text{ m/s} \).

---