Question

The equation of a progressive wave is given by \(y = 6cos(100t-4x)\) where \(y\) is in \(μ\text{m}\), \(x\) in metre and \(t\) in second. The ratio of the maximum particle velocity to the velocity of wave is

• A. \(1.5\times 10^{-5}\)
• B. \(2.0\times 10^{-6}\)
• C. \(2.4\times 10^{-5}\)
• D. \(2.8\times 10^{-6}\)

Hint:

Maximum particle velocity is A times omega and the wave velocity is omega over k.

Answer 1

The Correct Option is C

Step 1: Understand the concept
For \(y = A\cos(\omega t - kx)\), the maximum particle velocity is \(v_{p,max} = A\omega\), and the wave speed is \(v = \dfrac{\omega}{k}\).

Step 2: Read the data
\(A = 6\ \mu\text{m} = 6\times10^{-6}\) m, \(\omega = 100\) rad/s and \(k = 4\ \text{m}^{-1}\).

Step 3: Compute
\(v_{p,max} = 6\times10^{-6}\times100 = 6\times10^{-4}\) m/s. \(v = \dfrac{100}{4} = 25\) m/s.

Step 4: Ratio
\[ \frac{v_{p,max}}{v} = \frac{6\times10^{-4}}{25} = 2.4\times10^{-5} \]
Option (C).

Final Answer:
The ratio is 2.4 x 10^-5. This is option (C). \[ \boxed{\text{(C) }2.4\times10^{-5}} \]