A splash is heard 3.91 after a stone is dropped into a well of 67.6 m deep. The speed of sound in air is

Options

A. 346m/s
B. 320m/s
C. 330m/s
D. 338m/s

Answer

To solve the problem, we need to determine the speed of sound in air based on the time it takes for a stone to fall into a well and the subsequent splash sound to be heard. Here's a step-by-step solution:

### Step 1: Understand the Problem
The total time from when the stone is dropped until the splash sound is heard is given as 3.91 seconds. The depth of the well is 67.6 meters. The stone falls freely under gravity, and then the sound travels back up the well.

### Step 2: Identify the Known Values
- Depth of the well (s) = 67.6 m
- Total time (T) = 3.91 s
- Acceleration due to gravity (g) = 9.8 m/s² (we will use this value for calculations)

### Step 3: Calculate the Time of Free Fall
The stone is dropped from rest, so its initial velocity (u) is 0. We can use the second equation of motion:
\[ s = ut + \frac{1}{2} a t^2 \]
Substituting the known values:
\[ 67.6 = 0 \cdot t + \frac{1}{2} \cdot 9.8 \cdot t^2 \]
This simplifies to:
\[ 67.6 = 4.9 t^2 \]

### Step 4: Solve for Time of Free Fall (t)
Rearranging the equation gives:
\[ t^2 = \frac{67.6}{4.9} \]
Calculating this:
\[ t^2 = 13.8 \]
Taking the square root:
\[ t = \sqrt{13.8} \approx 3.71 \text{ seconds} \]

### Step 5: Calculate the Time for Sound to Travel
Now, we know the total time (T) is 3.91 seconds, and the time taken for the stone to fall (t) is approximately 3.71 seconds. The time taken for the sound to travel back up the well (t1) can be calculated as:
\[ t1 = T - t \]
\[ t1 = 3.91 - 3.71 = 0.20 \text{ seconds} \]

### Step 6: Calculate the Speed of Sound
The speed of sound (v) can be calculated using the formula:
\[ v = \frac{\text{distance}}{\text{time}} \]
The distance is the depth of the well (67.6 m), and the time is the time for sound to travel back up (0.20 s):
\[ v = \frac{67.6}{0.20} \]
Calculating this gives:
\[ v = 338 \text{ m/s} \]

### Final Answer
The speed of sound in air is approximately **338 m/s**.

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