Two waves of wave length 1m and 1.01 m produces 10 beats in 3 seconds the velocity of a gas about

To find the velocity of the gas based on the information about the two waves, we can start by analyzing the given data. We know the wavelengths of the two waves and the number of beats produced over a specific time period. Let's break this down step by step.

Understanding Beats and Wave Properties

When two waves of slightly different frequencies interfere with each other, they produce a phenomenon known as beats. The beat frequency is equal to the absolute difference between the frequencies of the two waves. The formula for beat frequency (f_beat) can be expressed as:

• f_beat = |f1 - f2|

Where f1 and f2 are the frequencies of the two waves. The frequency of a wave can be calculated using the formula:

• f = v / λ

Here, v represents the velocity of the wave, and λ is the wavelength. In this case, we have two wavelengths:

• λ1 = 1 m
• λ2 = 1.01 m

Calculating the Frequencies

Let's denote the velocities of the waves as v. The frequencies of the two waves can be expressed as:

• f1 = v / λ1 = v / 1
• f2 = v / λ2 = v / 1.01

Now, substituting these into the beat frequency formula gives us:

• f_beat = |(v / 1) - (v / 1.01)|

Finding the Beat Frequency

We know that the two waves produce 10 beats in 3 seconds, which means the beat frequency is:

• f_beat = 10 beats / 3 seconds = 3.33 Hz

Setting Up the Equation

Now we can set the two expressions for the beat frequency equal to each other:

• 3.33 = |(v / 1) - (v / 1.01)|

To simplify this, we can express it as:

• 3.33 = v(1 - 1/1.01)

Calculating the term in parentheses:

• 1 - 1/1.01 = 1 - 0.9901 = 0.0099

Solving for Velocity

Now we can substitute this back into our equation:

• 3.33 = v * 0.0099

To find v, we rearrange the equation:

• v = 3.33 / 0.0099

Calculating this gives:

• v ≈ 336.36 m/s

Final Thoughts

The velocity of the gas, based on the information about the two waves and the beats produced, is approximately 336.36 m/s. This calculation illustrates how wave properties and interference can be used to derive important physical quantities in wave mechanics.