To solve this problem, we will use the concept of beats and the Doppler effect.
Given:
Frequency of sound waves produced by both sources S1 and S2: f = 660 Hz
Speed of sound in air: v = 330 m/s
The listener hears 10 beats per second: beat frequency = 10 Hz
The listener is moving towards source S2 with speed u m/s.
Step 1: Understanding the Beat Frequency
The beat frequency is caused by the interference between the sound waves from two sources. The beat frequency is given by:
Beat frequency = |f'1 - f'2|
Where:
f'1 is the frequency heard by the listener from source S1
f'2 is the frequency heard by the listener from source S2
Step 2: Doppler Effect for Moving Listener
For a listener moving towards a sound source, the observed frequency increases according to the Doppler effect formula:
f' = f * (v / (v - u))
Where:
f is the frequency of the source
v is the speed of sound
u is the speed of the listener moving towards the source
For source S1 (listener moving towards S1), the frequency heard is:
f'1 = 660 * (330 / (330 - u))
For source S2 (listener moving towards S2), the frequency heard is:
f'2 = 660 * (330 / (330 + u))
Step 3: Calculate the Beat Frequency
The beat frequency is the absolute difference between f'1 and f'2:
Beat frequency = |f'1 - f'2|
Substitute the expressions for f'1 and f'2:
10 = |660 * (330 / (330 - u)) - 660 * (330 / (330 + u))|
Simplifying:
10 = 660 * |(330 / (330 - u)) - (330 / (330 + u))|
Factor out 660:
10 = 660 * |((330 + u) - (330 - u)) / ((330 - u)(330 + u))|
Simplify the numerator:
10 = 660 * |(2u) / (330^2 - u^2)|
Simplify further:
10 = 660 * (2u) / (108900 - u^2)
Step 4: Solve for u
Now, we solve for u. First, multiply both sides by (108900 - u^2):
10 * (108900 - u^2) = 1320u
Expand:
1089000 - 10u^2 = 1320u
Rearrange the equation:
10u^2 + 1320u - 1089000 = 0
Solve this quadratic equation using the quadratic formula:
u = (-b ± √(b^2 - 4ac)) / 2a
Here, a = 10, b = 1320, and c = -1089000. Substituting these values into the formula:
u = (-1320 ± √(1320^2 - 4 * 10 * -1089000)) / (2 * 10)
u = (-1320 ± √(1742400 + 43560000)) / 20
u = (-1320 ± √45302400) / 20
u = (-1320 ± 6737.4) / 20
Now, we have two possible solutions:
u = (-1320 + 6737.4) / 20 = 5417.4 / 20 = 270.87 m/s (which is not reasonable for this problem)
u = (-1320 - 6737.4) / 20 = -8057.4 / 20 = -402.87 m/s (this is also not reasonable)
Step 5: Recheck the options and process
