In a two dimensional motion,instantaneous speed `v_(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 \( v_0 \) in a two-dimensional motion. Let's break down the options given in the question step by step.

### Step-by-Step Solution:

1. **Understanding Instantaneous Speed**:
- Instantaneous speed being a positive constant \( v_0 \) means that the magnitude of the velocity does not change over time. However, this does not imply that the direction of the velocity is constant.

2. **Analyzing the Options**:
- **Option 1: The acceleration of the particle is zero**.
- This statement is **false**. A constant speed does not mean that there is no acceleration. Acceleration can still exist if the direction of the velocity vector changes, even if its magnitude remains constant.

- **Option 2: The acceleration of the particle is bounded**.
- This statement is also **false**. We do not have enough information to ascertain whether the acceleration is bounded or unbounded. The speed being constant does not provide any information about the limits of acceleration.

- **Option 3: The acceleration of the particle is necessarily in the plane of the motion**.
- This statement is **true**. Since the motion is confined to two dimensions, the acceleration must also lie within that same plane. This is a fundamental property of two-dimensional motion.

- **Option 4: The particle must be undergoing uniform circular motion**.
- This statement is **false**. While uniform circular motion does involve constant speed, the particle could also be moving in a straight line with constant speed. Therefore, we cannot conclude that the motion is circular based solely on the information given.

3. **Conclusion**:
- From the analysis above, the only necessarily true statement is that the acceleration of the particle is necessarily in the plane of the motion.

### Final Answer:
The only necessarily true statement is that **the acceleration of the particle is necessarily in the plane of the motion**.