Question

The velocity of sound in a gas in which waves of wavelength \(1\ \text{m}\) and \(1.01\ \text{m}\) produces \(10\) beats per second is:

• A. \(505\ \text{m/s}\)
• B. \(100\ \text{m/s}\)
• C. \(1010\ \text{m/s}\)
• D. \(101\ \text{m/s}\)

Hint:

For beats, use \(n=|f_1-f_2|\) and replace frequency by \(f=\frac{v}{\lambda}\).

Answer 1

The Correct Option is C

Concept:
Beat frequency is equal to the difference between two frequencies. \[ n=|f_1-f_2| \] Also, \[ f=\frac{v}{\lambda} \]

Step 1: Write the given wavelengths. \[ \lambda_1=1\ \text{m} \] \[ \lambda_2=1.01\ \text{m} \] Beat frequency: \[ n=10\ \text{s}^{-1} \]

Step 2: Write frequencies. \[ f_1=\frac{v}{1} \] \[ f_2=\frac{v}{1.01} \]

Step 3: Apply beat frequency formula. \[ 10=\frac{v}{1}-\frac{v}{1.01} \] \[ 10=v\left(1-\frac{1}{1.01}\right) \] \[ 10=v\left(\frac{1.01-1}{1.01}\right) \] \[ 10=v\left(\frac{0.01}{1.01}\right) \] \[ v=10\times \frac{1.01}{0.01} \] \[ v=1010\ \text{m/s} \] \[ \therefore \text{Correct Answer is (C)} \]