A stone is dropped into a well 44.1 deep. The sound of splash is heard 0 . 13 seconds after the stone hits the water. What should be the velocity of sounds in air .

Options

A. 319 m/s
B. 339 m/s
C. 359 m/s
D. 369m/s

Explanation

To solve the problem, we need to find the velocity of sound in air based on the information provided about the stone dropped into the well.

### Step-by-Step Solution:

1. **Identify the given data:**
- Depth of the well (H) = 44.1 m
- Total time taken to hear the splash (T) = 0.13 s

2. **Understand the events:**
- When the stone is dropped, it takes some time (t1) to reach the water.
- After the stone hits the water, sound travels back up to the top of the well, taking some time (t2).
- The total time is the sum of these two times:
\[ T = t1 + t2 \]

3. **Calculate the time taken for the stone to fall (t1):**
- The time taken for the stone to fall can be calculated using the formula for free fall:
\[ H = \frac{1}{2} g t1^2 \]
- Here, \( g \) (acceleration due to gravity) is approximately \( 9.81 \, \text{m/s}^2 \).
- Rearranging the formula gives:
\[ t1 = \sqrt{\frac{2H}{g}} = \sqrt{\frac{2 \times 44.1}{9.81}} \]

4. **Calculate t1:**
- Plugging in the values:
\[ t1 = \sqrt{\frac{88.2}{9.81}} \approx \sqrt{8.99} \approx 3.0 \, \text{s} \]

5. **Calculate the time taken for sound to travel back up (t2):**
- Since \( T = t1 + t2 \), we can find \( t2 \):
\[ t2 = T - t1 = 0.13 - t1 \]
- Substituting \( t1 \):
\[ t2 = 0.13 - 3.0 \approx -2.87 \, \text{s} \]
- Since this value is negative, we need to re-evaluate the approach.

6. **Correctly calculate the time taken for sound to travel back up:**
- The time taken for sound to travel back up can be calculated using the formula:
\[ t2 = \frac{H}{v} \]
- Where \( v \) is the velocity of sound in air.

7. **Combine the equations:**
- We know:
\[ T = t1 + t2 \]
- Rearranging gives:
\[ t2 = T - t1 \]
- Therefore:
\[ \frac{H}{v} = T - t1 \]

8. **Substituting values:**
- Rearranging for \( v \):
\[ v = \frac{H}{T - t1} \]

9. **Final calculation:**
- Substitute \( H = 44.1 \, \text{m} \) and \( T = 0.13 \, \text{s} \):
\[ v = \frac{44.1}{0.13 - t1} \]
- Calculate \( v \) to find the velocity of sound.