A train moves towards a stationary observer with speed `34 m//s`. The train sounds a whistle and its frequency registered by the observer is `f_(1)`. If the train's speed is reduced to `17 m//s`, the frequency registered is `f_(2)`. If the speed of sound of `340 m//s`, then the ratio `f_(1)//f_(2)`is

Options

A. `(18)/(19)`
B. `(1)/(2)`
C. 2
D. `(19)/(18)`

Answer

To solve the problem, we will use the formula for the apparent frequency heard by a stationary observer when a source of sound is moving towards them. The formula is given by:

\[ f' = f_0 \frac{v}{v - v_s} \]

where:
- \( f' \) is the apparent frequency heard by the observer,
- \( f_0 \) is the actual frequency of the source,
- \( v \) is the speed of sound,
- \( v_s \) is the speed of the source.

### Step 1: Calculate \( f_1 \) when the train's speed is 34 m/s

For the first case, where the speed of the train \( v_s = 34 \, \text{m/s} \):

\[ f_1 = f_0 \frac{v}{v - v_s} = f_0 \frac{340}{340 - 34} \]

Calculating the denominator:

\[ 340 - 34 = 306 \]

Thus, we have:

\[ f_1 = f_0 \frac{340}{306} \]

### Step 2: Calculate \( f_2 \) when the train's speed is 17 m/s

For the second case, where the speed of the train \( v_s = 17 \, \text{m/s} \):

\[ f_2 = f_0 \frac{v}{v - v_s} = f_0 \frac{340}{340 - 17} \]

Calculating the denominator:

\[ 340 - 17 = 323 \]

Thus, we have:

\[ f_2 = f_0 \frac{340}{323} \]

### Step 3: Find the ratio \( \frac{f_1}{f_2} \)

Now, we can find the ratio of the two frequencies:

\[ \frac{f_1}{f_2} = \frac{f_0 \frac{340}{306}}{f_0 \frac{340}{323}} = \frac{340}{306} \cdot \frac{323}{340} \]

The \( f_0 \) and \( 340 \) cancel out:

\[ \frac{f_1}{f_2} = \frac{323}{306} \]

### Step 4: Simplify the ratio

Now we simplify \( \frac{323}{306} \):

To get the simplest form, we can divide both the numerator and denominator by their greatest common divisor (GCD). In this case, both numbers do not have a common factor other than 1, so we can express it as:

\[ \frac{f_1}{f_2} = \frac{323}{306} \approx \frac{19}{18} \]

### Final Answer

Thus, the ratio \( \frac{f_1}{f_2} \) is:

\[ \frac{f_1}{f_2} = \frac{19}{18} \]