A body moves in anticlockwise direction on a circular path in the x-y plane. The radius of the circular path is 5m and its centre is at the origin. In a certain interval of time, displacement of the body is obesrved to be 6m in the positive y-direction. Which of the following is true ?

Options

A. Its initial position vector is `5hati` m
B. Its initial position vector is `(-3hati+4hatj)` m.
C. Its final position vector is `(4hati+3hatj)`m.
D. Its final position vector is `6hatj` m.

Answer

To solve the problem, we need to analyze the motion of the body moving in an anticlockwise direction on a circular path with a radius of 5 m. The displacement of the body is given as 6 m in the positive y-direction.

### Step-by-Step Solution:

1. **Understanding the Circular Motion**:
- The body moves in a circular path with a radius of 5 m, centered at the origin (0,0) in the x-y plane.
- The position of the body can be represented using the angle θ made with the x-axis.

2. **Position Vector Representation**:
- The position vector \( \vec{r} \) of the body at any angle θ can be expressed as:
\[ \vec{r} = 5 \cos(\theta) \hat{i} + 5 \sin(\theta) \hat{j} \]

3. **Displacement Vector**:
- The displacement is given as 6 m in the positive y-direction, which can be represented as:
\[ \Delta \vec{r} = 0 \hat{i} + 6 \hat{j} \]

4. **Initial and Final Position Vectors**:
- Let the initial position vector be \( \vec{r}_{initial} = 5 \cos(\theta_{initial}) \hat{i} + 5 \sin(\theta_{initial}) \hat{j} \)
- Let the final position vector be \( \vec{r}_{final} = 5 \cos(\theta_{final}) \hat{i} + 5 \sin(\theta_{final}) \hat{j} \)

5. **Calculating the Change in Position Vector**:
- The change in position vector (displacement) is given by:
\[ \Delta \vec{r} = \vec{r}_{final} - \vec{r}_{initial} \]
- This can be expressed as:
\[ \Delta \vec{r} = (5 \cos(\theta_{final}) - 5 \cos(\theta_{initial})) \hat{i} + (5 \sin(\theta_{final}) - 5 \sin(\theta_{initial})) \hat{j} \]

6. **Setting Up the Equations**:
- From the displacement vector, we know:
\[ 0 \hat{i} + 6 \hat{j} = (5 \cos(\theta_{final}) - 5 \cos(\theta_{initial})) \hat{i} + (5 \sin(\theta_{final}) - 5 \sin(\theta_{initial})) \hat{j} \]
- This gives us two equations:
1. \( 5 \cos(\theta_{final}) - 5 \cos(\theta_{initial}) = 0 \)
2. \( 5 \sin(\theta_{final}) - 5 \sin(\theta_{initial}) = 6 \)

7. **Solving the Equations**:
- From the first equation, we can simplify:
\[ \cos(\theta_{final}) = \cos(\theta_{initial}) \]
- From the second equation:
\[ \sin(\theta_{final}) - \sin(\theta_{initial}) = \frac{6}{5} \]

8. **Using the Sine and Cosine Values**:
- The maximum value of \( \sin \) is 1, which means the change in sine must be valid within the range of the sine function. This indicates that the angles must be such that:
\[ \sin(\theta_{final}) = \sin(\theta_{initial}) + \frac{6}{5} \]
- This is impossible since the sine function cannot exceed 1. Therefore, the values of \( \theta_{initial} \) and \( \theta_{final} \) must be such that they yield a valid sine value.

9. **Conclusion**:
- The only valid conclusion is that the body must have moved in a way that maintains the circular path while achieving the given displacement in the y-direction. The angles must be such that the cosine values are equal, and the sine values differ by a maximum of 1.