A particle is moving in a plane with velocity given by `vec(u)=u_(0)hat(i)+(aomega cos omegat)hat(j)`, where `hat(i)` and `hat(j)` are unit vectors along x and y axes respectively. If particle is at the origin at `t=0`. Calculate the trajectory of the particle :-

Options

A. `y= a sin (u_(0)/(omega x))`
B. `y= asin ((omega x)/u_(0))`
C. `y=1/a. sin (u_(0)/(omega x))`
D. `y=1/a. sin((omega x)/u_(0))`

Explanation

To find the trajectory of the particle given its velocity vector, we will integrate the velocity components with respect to time.

### Step-by-Step Solution:

1. **Identify the velocity components**:
The velocity of the particle is given as:
\[ \vec{u} = u_0 \hat{i} + a \omega \cos(\omega t) \hat{j} \]
This means:
- The x-component of velocity, \( u_x = u_0 \)
- The y-component of velocity, \( u_y = a \omega \cos(\omega t) \)

2. **Integrate the x-component**:
To find the position in the x-direction, we integrate the x-component of velocity:
\[ x(t) = \int u_x \, dt = \int u_0 \, dt = u_0 t + C_x \]
Since the particle is at the origin when \( t = 0 \) (i.e., \( x(0) = 0 \)), we find:
\[ C_x = 0 \quad \Rightarrow \quad x(t) = u_0 t \]

3. **Integrate the y-component**:
Now, we integrate the y-component of velocity:
\[ y(t) = \int u_y \, dt = \int a \omega \cos(\omega t) \, dt \]
The integral of \( \cos(\omega t) \) is:
\[ y(t) = a \omega \left( \frac{\sin(\omega t)}{\omega} \right) + C_y = a \sin(\omega t) + C_y \]
Again, since the particle is at the origin when \( t = 0 \) (i.e., \( y(0) = 0 \)), we find:
\[ C_y = 0 \quad \Rightarrow \quad y(t) = a \sin(\omega t) \]

4. **Substitute \( t \) from \( x(t) \)**:
From the equation \( x(t) = u_0 t \), we can express \( t \) in terms of \( x \):
\[ t = \frac{x}{u_0} \]
Now substitute this expression for \( t \) into the equation for \( y(t) \):
\[ y = a \sin\left(\omega \frac{x}{u_0}\right) \]

5. **Final trajectory equation**:
The trajectory of the particle is given by:
\[ y = a \sin\left(\frac{\omega}{u_0} x\right) \]

### Conclusion:
The trajectory of the particle is:
\[ y = a \sin\left(\frac{\omega}{u_0} x\right) \]