A stone is dropped into a pond from the top of the tower of height h. If v is the speed of sound in air, then the sound of splash will be heard at the top of the tower after a time

Options

A. `sqrt((2h)/g)+h/v`
B. `sqrt((2h)/g)-h/v`
C. `sqrt((2h)/g)`
D. `sqrt((2h)/g)+(2h)/v`

Explanation

To solve the problem of determining the total time taken for the sound of the splash to be heard at the top of the tower after a stone is dropped into a pond, we can break it down into two parts: the time taken for the stone to fall to the pond (T1) and the time taken for the sound to travel back up to the top of the tower (T2).

### Step-by-step Solution:

1. **Identify the Variables**:
- Let \( h \) be the height of the tower.
- Let \( g \) be the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)).
- Let \( v \) be the speed of sound in air.

2. **Calculate the Time for the Stone to Fall (T1)**:
- The stone is dropped from rest, so its initial velocity \( u = 0 \).
- We can use the equation of motion for free fall:
\[ h = ut + \frac{1}{2} g t_1^2 \]
Substituting \( u = 0 \):
\[ h = \frac{1}{2} g t_1^2 \]
- Rearranging gives:
\[ t_1^2 = \frac{2h}{g} \]
- Taking the square root:
\[ t_1 = \sqrt{\frac{2h}{g}} \]

3. **Calculate the Time for Sound to Travel Back Up (T2)**:
- The time taken for the sound to travel back up to the top of the tower is given by:
\[ t_2 = \frac{h}{v} \]

4. **Calculate the Total Time (T)**:
- The total time \( T \) is the sum of \( T1 \) and \( T2 \):
\[ T = t_1 + t_2 \]
- Substituting the values we found:
\[ T = \sqrt{\frac{2h}{g}} + \frac{h}{v} \]

### Final Answer:
The total time taken for the sound of the splash to be heard at the top of the tower is:
\[ T = \sqrt{\frac{2h}{g}} + \frac{h}{v} \]