A car moving towards an approaching bike. The bike is moving with a velocity `30ms^(-1)`. Frequency of horn sounded by the car is 100 Hz, while it is observed by bike rider as 120 Hz. The actual duration of horn is 6 s. Find the time interval for which the bike rider hears it. (Take velocity of sound in air as `330ms^(-1)`)

Options

A. 5s
B. 6s
C. 7s
D. Cannot be calcualted with given information

Answer

To solve the problem, we need to find the time interval for which the bike rider hears the horn of the car. We will use the concept of apparent frequency and the relationship between distance, speed, and time.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Frequency of the horn (fs) = 100 Hz
- Observed frequency (fo) = 120 Hz
- Velocity of the bike (vb) = 30 m/s (towards the car)
- Velocity of sound in air (v) = 330 m/s
- Actual duration of the horn (t) = 6 s

2. **Use the Formula for Apparent Frequency:**
The formula for apparent frequency when both the source and observer are moving is given by:
\[ f_o = \frac{v + v_o}{v - v_s} f_s \]
Where:
- \( f_o \) = observed frequency
- \( f_s \) = source frequency
- \( v \) = speed of sound in air
- \( v_o \) = speed of the observer (bike)
- \( v_s \) = speed of the source (car)

Here, since the bike is moving towards the car, we take \( v_o \) as positive. We need to find \( v_s \) (the speed of the car).

3. **Substituting Known Values:**
Substituting the known values into the formula:
\[ 120 = \frac{330 + 30}{330 - v_s} \cdot 100 \]

4. **Simplifying the Equation:**
Rearranging the equation to isolate \( v_s \):
\[ 120(330 - v_s) = 100(360) \]
\[ 120 \cdot 330 - 120 v_s = 36000 \]
\[ 39600 - 120 v_s = 36000 \]
\[ 120 v_s = 39600 - 36000 \]
\[ 120 v_s = 3600 \]
\[ v_s = \frac{3600}{120} = 30 \text{ m/s} \]

5. **Calculate the Distance Traveled by Sound:**
The distance traveled by the sound during the time the horn is sounded is:
\[ d = v \cdot t = 330 \cdot 6 = 1980 \text{ m} \]

6. **Calculate the Time for Which the Biker Hears the Horn:**
The distance traveled by the sound can also be expressed in terms of the time \( T \) the biker hears the sound:
\[ d = v \cdot T \]
The biker is moving towards the car at 30 m/s, and the sound is also moving towards him at 330 m/s. The effective speed of sound relative to the biker is:
\[ v_{effective} = v + v_b = 330 + 30 = 360 \text{ m/s} \]
Now, we can find the time \( T \):
\[ 1980 = 360 \cdot T \]
\[ T = \frac{1980}{360} = 5.5 \text{ seconds} \]

### Final Answer:
The time interval for which the bike rider hears the horn is **5.5 seconds**.