A stone is dropped from roof of building takes 5 s to reach the ground. The sound of striking is heard after 5.4 second after the stone is dropped. The velocity of sound is

Options

A. `312.5 ms^(-1)`
B. `332.5 ms^(-1)`
C. `352 ms^(-1)`
D. `392 ms^(-1)`

Answer

To solve the problem of finding the velocity of sound given the time taken for a stone to fall and the time taken for the sound to reach the observer, we can break it down into the following steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
- A stone is dropped from the roof of a building and takes 5 seconds to reach the ground.
- The sound of the stone striking the ground is heard 5.4 seconds after the stone is dropped.

2. **Calculate the Height of the Building**:
- We will use the second equation of motion to find the height (H) of the building. The equation is:
\[ S = ut + \frac{1}{2} a t^2 \]
- Here, \( S \) is the displacement (height of the building), \( u \) is the initial velocity (0 m/s since the stone is dropped), \( a \) is the acceleration due to gravity (approximately \( 10 \, \text{m/s}^2 \)), and \( t \) is the time taken (5 seconds).
- Substituting the values:
\[ H = 0 \times 5 + \frac{1}{2} \times 10 \times (5)^2 \]
\[ H = 0 + \frac{1}{2} \times 10 \times 25 \]
\[ H = 125 \, \text{meters} \]

3. **Determine the Time Taken for Sound to Travel**:
- The total time from dropping the stone to hearing the sound is 5.4 seconds.
- The time taken for the sound to travel back to the observer after the stone hits the ground is:
\[ \text{Time for sound} = 5.4 \, \text{s} - 5 \, \text{s} = 0.4 \, \text{s} \]

4. **Calculate the Velocity of Sound**:
- The velocity of sound can be calculated using the formula:
\[ v = \frac{d}{t} \]
- Where \( d \) is the distance (height of the building, 125 meters) and \( t \) is the time taken for the sound to travel (0.4 seconds).
- Substituting the values:
\[ v = \frac{125 \, \text{meters}}{0.4 \, \text{seconds}} = 312.5 \, \text{m/s} \]

5. **Final Answer**:
- The velocity of sound is \( 312.5 \, \text{m/s} \).