Question

The equation of a plane progressive wave is given by} \[ y = 5\cos\pi\left(200t - \frac{x}{150}\right) \] where \(x\) and \(y\) are in cm and \(t\) is in seconds. The velocity of the wave is ____ m/s.

• A. \(120\)
• B. \(150\)
• C. \(200\)
• D. \(300\)

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A plane progressive wave is described by the equation \(y = A \cos(\omega t - kx)\). The wave velocity (\(v\)) is determined by the ratio of the angular frequency (\(\omega\)) to the wave number (\(k\)).
Step 2: Key Formula or Approach:
1. Standard form: \(y = A \cos(\omega t - kx)\)
2. Wave velocity: \(v = \frac{\omega}{k}\)
Step 3: Detailed Explanation:
The given equation is:
\[ y = 5 \cos \left( 200 \pi t - \frac{\pi x}{150} \right) \]
Comparing with the standard form \(y = A \cos(\omega t - kx)\):
Angular frequency \(\omega = 200\pi \text{ rad/s}\)
Wave number \(k = \frac{\pi}{150} \text{ rad/cm}\)
Calculate wave velocity \(v\):
\[ v = \frac{\omega}{k} = \frac{200\pi}{\pi / 150} = 200 \times 150 = 30,000 \text{ cm/s} \]
The question asks for the velocity in \text{m/s}. Convert \text{cm/s} to \text{m/s}:
\[ v = \frac{30,000}{100} \text{ m/s} = 300 \text{ m/s} \]
Step 4: Final Answer:
The velocity of the wave is 300 m/s.