A splash in heard `3.12` s after a stone is dropped into a well 45 m deep. The speed of sound in air is `[ g =10 ms ^(-2)]`

Options

A. `330 ms ^(-1)`
B. `375ms ^(-1)`
C. `340ms ^(-1)`
D. `346ms ^(-1)`

Answer

To solve the problem, we need to break it down into steps:

### Step 1: Understand the problem
We are given that a stone is dropped into a well that is 45 meters deep, and the splash is heard 3.12 seconds later. We need to find the speed of sound in air.

### Step 2: Calculate the time taken by the stone to reach the water surface
The time taken by the stone to fall to the water surface can be calculated using the formula for free fall:
\[ t_1 = \sqrt{\frac{2h}{g}} \]
where:
- \( h = 45 \) m (depth of the well)
- \( g = 10 \, \text{m/s}^2 \) (acceleration due to gravity)

Substituting the values:
\[ t_1 = \sqrt{\frac{2 \times 45}{10}} \]
\[ t_1 = \sqrt{\frac{90}{10}} \]
\[ t_1 = \sqrt{9} \]
\[ t_1 = 3 \, \text{s} \]

### Step 3: Calculate the time taken for sound to travel back up
The total time from when the stone is dropped to when the splash is heard is 3.12 seconds. Therefore, the time taken by the sound to travel back up to the top of the well is:
\[ t_{sound} = \text{Total time} - t_1 \]
\[ t_{sound} = 3.12 \, \text{s} - 3 \, \text{s} \]
\[ t_{sound} = 0.12 \, \text{s} \]

### Step 4: Calculate the speed of sound
The speed of sound can be calculated using the formula:
\[ \text{Speed of sound} = \frac{h}{t_{sound}} \]
Substituting the values:
\[ \text{Speed of sound} = \frac{45 \, \text{m}}{0.12 \, \text{s}} \]
\[ \text{Speed of sound} = 375 \, \text{m/s} \]

### Conclusion
The speed of sound in air is \( 375 \, \text{m/s} \).