A stone is dropped into a lake from a tower 500 metre high. The sound of the splash will be heard by the man approximately after

Options

A. 11.5 seconds
B. 21 seconds
C. 10 seconds
D. 14 seconds

Answer

To solve the problem of how long it takes for a man to hear the sound of a splash after dropping a stone from a height of 500 meters, we need to calculate two times: the time it takes for the stone to fall (T1) and the time it takes for the sound to travel back up to the man (T2).

### Step-by-step Solution:

1. **Calculate the time for the stone to fall (T1)**:
- We can use the equation of motion for free fall:
\[ S = ut + \frac{1}{2}gt^2 \]
- Here, \(S = 500 \, \text{m}\) (the height of the tower), \(u = 0 \, \text{m/s}\) (initial velocity since the stone is dropped), and \(g = 9.8 \, \text{m/s}^2\) (acceleration due to gravity).
- Plugging in the values:
\[ 500 = 0 \cdot t + \frac{1}{2} \cdot 9.8 \cdot t^2 \]
\[ 500 = 4.9t^2 \]
\[ t^2 = \frac{500}{4.9} \approx 102.04 \]
\[ t \approx \sqrt{102.04} \approx 10.1 \, \text{s} \]
- Therefore, the time taken for the stone to fall (T1) is approximately **10.1 seconds**.

2. **Calculate the time for the sound to travel back up (T2)**:
- The speed of sound in air is approximately \(v = 340 \, \text{m/s}\).
- The distance the sound travels is also \(500 \, \text{m}\).
- We can use the formula:
\[ T2 = \frac{d}{v} \]
- Plugging in the values:
\[ T2 = \frac{500}{340} \approx 1.47 \, \text{s} \]
- Therefore, the time taken for the sound to travel back up (T2) is approximately **1.47 seconds**.

3. **Calculate the total time (T)**:
- The total time for the man to hear the splash sound is the sum of T1 and T2:
\[ T = T1 + T2 \approx 10.1 + 1.47 \approx 11.57 \, \text{s} \]
- Rounding this, we can say the total time is approximately **11.5 seconds**.

### Final Answer:
The sound of the splash will be heard by the man approximately after **11.5 seconds**.