A wave `y = A cos(omega t – kx)` passes through a medium. If V is the particle velocity and a is the particle acceleration then,

Options

A. y,V and 'a’ all are in the same phase
B. y lags behind V by a phase angle of ` pi/2`
C. 'a' leads y by a phase angle of `pi`
D. 'V' leads a by a phase angle of `(3pi )/(2)`

Answer

To solve the problem, we need to analyze the wave function given by \( y = A \cos(\omega t - kx) \) and find the relationships between the wave displacement \( y \), particle velocity \( v \), and particle acceleration \( a \).

### Step 1: Find the Particle Velocity \( v \)

The particle velocity \( v \) is defined as the time derivative of the displacement \( y \). Therefore, we differentiate \( y \) with respect to time \( t \):

\[ v = \frac{dy}{dt} = \frac{d}{dt} (A \cos(\omega t - kx)) \]

Using the chain rule, we get:

\[ v = -A \omega \sin(\omega t - kx) \]

This can be expressed in terms of cosine:

\[ v = A \omega \cos\left(\omega t - kx + \frac{\pi}{2}\right) \]

### Step 2: Find the Particle Acceleration \( a \)

The particle acceleration \( a \) is defined as the time derivative of the velocity \( v \). Therefore, we differentiate \( v \) with respect to time \( t \):

\[ a = \frac{dv}{dt} = \frac{d}{dt} (-A \omega \sin(\omega t - kx)) \]

Using the chain rule again, we get:

\[ a = -A \omega^2 \cos(\omega t - kx) \]

This can also be expressed in terms of cosine:

\[ a = A \omega^2 \cos\left(\omega t - kx + \pi\right) \]

### Step 3: Analyze the Phase Relationships

Now, we can analyze the phase relationships between \( y \), \( v \), and \( a \):

1. **Phase of \( y \)**: \( \phi_y = \omega t - kx \)
2. **Phase of \( v \)**: \( \phi_v = \omega t - kx + \frac{\pi}{2} \)
3. **Phase of \( a \)**: \( \phi_a = \omega t - kx + \pi \)

From this, we can conclude:

- \( y \) lags behind \( v \) by \( \frac{\pi}{2} \) (since \( \phi_v - \phi_y = \frac{\pi}{2} \)).
- \( a \) leads \( y \) by \( \pi \) (since \( \phi_a - \phi_y = \pi \)).
- \( v \) leads \( a \) by \( \frac{3\pi}{2} \) (since \( \phi_v - \phi_a = \frac{3\pi}{2} \)).

### Conclusion

Based on the analysis, we can summarize the relationships:

- \( y \) lags behind \( v \) by \( \frac{\pi}{2} \).
- \( a \) leads \( y \) by \( \pi \).
- \( v \) leads \( a \) by \( \frac{3\pi}{2} \).