In a two dimensional motion, instantaneous speed `v_(0)` is a positive constant. Then which of the following are neccessarily 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 question regarding the two-dimensional motion with a constant instantaneous speed \( v_0 \), we will analyze each of the given options step by step.

### Step-by-Step Solution:

1. **Understanding Instantaneous Speed**:
- Instantaneous speed \( v_0 \) is defined as the magnitude of the velocity of an object at a specific moment in time. It is a scalar quantity and does not provide any information about the direction of motion.

2. **Evaluating the Options**:
- We need to determine which of the given statements are necessarily true given that the instantaneous speed is a positive constant.

3. **Option 1: Average Velocity is Not Zero at Any Time**:
- Average velocity is defined as the total displacement divided by the total time taken. Since instantaneous speed is constant, the object can still return to its starting point, resulting in zero displacement. Thus, this statement is **not necessarily true**.

4. **Option 2: Average Acceleration Must Always Vanish**:
- Average acceleration is defined as the change in velocity divided by the time taken. Since speed is constant, the velocity may still change direction, leading to a non-zero average acceleration. Thus, this statement is **not necessarily true**.

5. **Option 3: Displacement in Equal Time Intervals are Equal**:
- Displacement is a vector quantity, and while speed is constant, the direction of motion can change. Therefore, the displacement in equal time intervals may not be equal if the path is curved. Thus, this statement is **not necessarily true**.

6. **Option 4: Equal Path Lengths are Transversed in Equal Time Intervals**:
- Since the instantaneous speed \( v_0 \) is constant, the distance traveled in equal time intervals will also be equal. This is because distance is calculated as speed multiplied by time. Thus, this statement is **necessarily true**.

### Conclusion:
The only statement that is necessarily true given the conditions of the problem is:
- **Equal path lengths are transversed in equal time intervals**.