In a two -dimensional motion, instantaneous speed `vecv_(0)` is a positive constant . Then which of the following are necessarily true ?

Options

A. The acceleration of the particle is zero
B. The acceleration of the particle is bounded
C. The acceleration of the particle is necessarily in the plane of motion
D. The particle must be undergoing a uniform circular motion .

Explanation

To solve the problem, we need to analyze the implications of having a constant instantaneous speed \( \vec{v}_0 \) in a two-dimensional motion.

### Step-by-Step Solution:

1. **Understanding Instantaneous Speed**:
- Instantaneous speed is defined as the magnitude of the velocity vector at a specific moment in time. If \( \vec{v}_0 \) is a positive constant, it means that the speed of the particle does not change over time.

2. **Implications of Constant Speed**:
- Since the speed is constant, the magnitude of the velocity vector remains the same. However, the direction of the velocity vector can change. This means that the particle can be moving in a curved path.

3. **Acceleration**:
- Acceleration is defined as the rate of change of velocity. If the speed is constant but the direction of the velocity vector is changing, then there must be an acceleration acting on the particle. This acceleration is responsible for changing the direction of the velocity vector while maintaining its magnitude.

4. **Direction of Acceleration**:
- For the acceleration to change the direction of the velocity while keeping its magnitude constant, the acceleration must be perpendicular to the velocity vector at all times. This is a key characteristic of uniform circular motion, but it can also apply to other types of curved motion.

5. **Boundedness of Acceleration**:
- The magnitude of the acceleration does not have to be constant or bounded. It can vary while still being perpendicular to the velocity vector. Therefore, the statement that the acceleration is bounded is not necessarily true.

6. **Plane of Motion**:
- Since the motion is two-dimensional and the acceleration is always perpendicular to the velocity, the motion must occur in a plane. However, it does not imply that the motion is circular; it can be any type of curve in that plane.

### Conclusion:
Based on the analysis, the only necessarily true statement is that the particle must be moving in a plane (option 3). The other statements regarding the nature of acceleration and its boundedness are not necessarily true.

### Summary of True Statements:
- The acceleration is always perpendicular to the velocity.
- The motion is in a plane.
- The acceleration is not necessarily bounded.