A turning fork vibrating at `500 Hz` falls from rest accelerates at `10 m//s^(2)`.
Time taken by the waves with a frequency of `475 Hz` to reach the release point is nearly

Options

A. `1.79 s`
B. `1.84 s`
C. `17.9 s`
D. `18.4 s`

Explanation

To solve the problem step by step, we need to analyze the situation involving the tuning fork and the sound waves. Here’s how we can break it down:

### Step 1: Understand the Problem
We have a tuning fork vibrating at a frequency of 500 Hz that falls from rest with an acceleration of 10 m/s². We need to find the time taken by sound waves of frequency 475 Hz to reach the release point of the tuning fork.

### Step 2: Use the Doppler Effect
The apparent frequency \( f' \) of the sound heard by an observer when the source is moving towards them can be given by the formula:
\[ f' = f_0 \frac{v + v_o}{v - v_s} \]
where:
- \( f' \) = apparent frequency (475 Hz)
- \( f_0 \) = original frequency (500 Hz)
- \( v \) = speed of sound in air (approximately 340 m/s)
- \( v_o \) = speed of the observer (0 m/s, since the observer is at rest)
- \( v_s \) = speed of the source (the tuning fork)

### Step 3: Set Up the Equation
Since the observer is at rest, we can simplify the equation:
\[ 475 = 500 \frac{340}{340 - v_s} \]

### Step 4: Solve for \( v_s \)
Rearranging the equation gives:
\[ 475(340 - v_s) = 500 \cdot 340 \]
Expanding and rearranging:
\[ 475 \cdot 340 - 475 v_s = 500 \cdot 340 \]
\[ -475 v_s = 500 \cdot 340 - 475 \cdot 340 \]
\[ -475 v_s = 25 \cdot 340 \]
\[ v_s = -\frac{25 \cdot 340}{475} \]
Calculating \( v_s \):
\[ v_s = -17.89 \text{ m/s} \quad (\text{taking the absolute value, } v_s \approx 17.9 \text{ m/s}) \]

### Step 5: Calculate the Time Taken to Fall
The tuning fork falls under gravity, so we can use the kinematic equation:
\[ v = u + at \]
where:
- \( u = 0 \) (initial velocity)
- \( a = 10 \text{ m/s}^2 \)
- \( v = v_s \approx 17.9 \text{ m/s} \)

Rearranging gives:
\[ t = \frac{v_s}{a} = \frac{17.9}{10} = 1.79 \text{ seconds} \]

### Step 6: Calculate the Distance Fallen
Using the formula for distance fallen:
\[ s = ut + \frac{1}{2} a t^2 \]
Since \( u = 0 \):
\[ s = \frac{1}{2} \cdot 10 \cdot (1.79)^2 \]
Calculating \( s \):
\[ s = 5 \cdot 3.2041 \approx 16.02 \text{ meters} \]

### Step 7: Calculate the Time for Sound to Travel Upwards
Now, we need to calculate the time taken by the sound to travel back up to the release point:
\[ t_{sound} = \frac{s}{v} = \frac{16.02}{340} \approx 0.047 \text{ seconds} \]

### Step 8: Total Time Calculation
The total time taken for the sound to reach the release point is:
\[ t_{total} = t + t_{sound} = 1.79 + 0.047 \approx 1.837 \text{ seconds} \]

### Final Answer
The time taken by the waves with a frequency of 475 Hz to reach the release point is approximately **1.837 seconds**.

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