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Question

A sound wave having frequency of 300 Hz is travelling in an unknown medium. Its wavelength is not known. It travels a distance equal to 150 times its wavelength in time t. The value of t is:

This question was previously asked in
CDS 2 2024 Maths Question Paper (01-Sep-2024)
The correct answer is
0.5 s

Sound Wave Travel Time Calculation

This problem asks us to find the time (\(t\)) it takes for a sound wave to travel a specific distance, given its frequency (\(f\)). We need to use the fundamental relationships between wave speed, frequency, wavelength, distance, and time.

Wave Properties Explained

Before calculating, let's recall the key properties of waves involved:

  • Frequency (\(f\)): This tells us how many complete wave cycles occur in one second. It's measured in Hertz (Hz). In this case, \(f = 300\) Hz.
  • Wavelength (\(\lambda\)): This is the length of one complete wave cycle, measured from crest to crest or trough to trough. It's measured in meters (m).
  • Speed (\(v\)): This is how fast the wave travels through a medium. It's measured in meters per second (m/s).
  • Distance (\(d\)): The total length the wave travels. Measured in meters (m).
  • Time (\(t\)): The duration of the travel. Measured in seconds (s).

Given Information

The problem provides us with:

  • The frequency of the sound wave: \(f = 300\) Hz.
  • The distance traveled (\(d\)) is given as 150 times the wavelength (\(\lambda\)): \(d = 150 \lambda\).

Travel Time Calculation

We need to determine the value of \(t\). We can use two main formulas for wave speed (\(v\)):

  1. The relationship between speed, frequency, and wavelength: \(v = f \lambda\)
  2. The relationship between speed, distance, and time: \(v = \frac{d}{t}\)

We are given that the distance \(d = 150 \lambda\). Let's substitute this into the second speed formula:

\(v = \frac{150 \lambda}{t}\)

Now, since the wave is traveling in a single medium, its speed (\(v\)) must be constant. Therefore, we can set the two expressions for \(v\) equal to each other:

\(f \lambda = \frac{150 \lambda}{t}\)

Notice that the wavelength (\(\lambda\)) appears on both sides of the equation. Since \(\lambda\) must be greater than zero for a wave to exist, we can cancel it out:

\(f = \frac{150}{t}\)

Our goal is to find the time \(t\). We can rearrange the equation to isolate \(t\):

\(t = \frac{150}{f}\)

Finally, we can substitute the given frequency value (\(f = 300\) Hz) into this equation:

\(t = \frac{150}{300 \text{ Hz}}\)

Calculating the value:

\(t = 0.5 \text{ s}\)

Conclusion

The calculation shows that the sound wave travels the specified distance in 0.5 seconds.

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