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

Concept: The general equation of a progressive wave is \[ y = A\cos(\omega t - kx) \] where \[ v=\frac{\omega}{k} \] is the wave velocity.
Step 1:Rewrite the given equation} \[ y = 5\cos\left(200\pi t - \frac{\pi x}{150}\right) \] Thus \[ \omega = 200\pi, \qquad k = \frac{\pi}{150} \]
Step 2:Find velocity} \[ v=\frac{\omega}{k} \] \[ v=\frac{200\pi}{\pi/150} \] \[ v=200 \times 150 \] \[ v=30000\ \text{cm/s} \]
Step 3:Convert to m/s} \[ 30000\ \text{cm/s} = 300\ \text{m/s} \] Thus \[ \boxed{300} \]