A stone is dropped in a well 107 m deep. If the splash is heard 5 seconds after the stone is dropped, then calculate the speed of sound in air.

Answer

To solve the problem of calculating the speed of sound in air given that a stone is dropped in a well 107 m deep and the splash is heard 5 seconds after the stone is dropped, we can follow these steps:

### Step 1: Understand the Problem
When the stone is dropped, it takes some time (T) to hit the water, and then the sound takes additional time (T') to travel back up the well. The total time for both events is given as 5 seconds.

### Step 2: Set Up the Equation
We can express the total time as:
\[ T + T' = 5 \text{ seconds} \]

### Step 3: Calculate the Time for the Stone to Fall (T)
Using the equation of motion for free fall:
\[ s = ut + \frac{1}{2} a t^2 \]
Where:
- \( s = 107 \) m (depth of the well)
- \( u = 0 \) (initial velocity, since the stone is dropped)
- \( a = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity)

Substituting the values:
\[ 107 = 0 \cdot T + \frac{1}{2} \cdot 9.8 \cdot T^2 \]
This simplifies to:
\[ 107 = 4.9 T^2 \]

### Step 4: Solve for T
Rearranging gives:
\[ T^2 = \frac{107}{4.9} \]
Calculating this:
\[ T^2 \approx 21.83 \]
Taking the square root:
\[ T \approx \sqrt{21.83} \approx 4.67 \text{ seconds} \]

### Step 5: Calculate the Time for Sound to Travel (T')
Using the total time equation:
\[ T' = 5 - T \]
Substituting the value of T:
\[ T' = 5 - 4.67 = 0.33 \text{ seconds} \]

### Step 6: Calculate the Speed of Sound
The speed of sound can be calculated using the formula:
\[ \text{Speed of Sound} = \frac{\text{Distance}}{\text{Time}} \]
The distance is the same as the depth of the well (107 m), and the time for the sound to travel back is T':
\[ \text{Speed of Sound} = \frac{107 \text{ m}}{0.33 \text{ s}} \]
Calculating this gives:
\[ \text{Speed of Sound} \approx 324.24 \text{ m/s} \]

### Final Answer
The speed of sound in air is approximately **324.24 m/s**.

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