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

Options

A. The average velocity is not zero at any time
B. Average acceleration must always vanish
C. Displacements in equal time intervals are equal
D. Equal path lengths are traversed in equal intervals

Explanation

To solve the question regarding the two-dimensional motion where the instantaneous speed \( v_0 \) is a positive constant, we need to analyze the given options one by one to determine which statements are necessarily true.

### Step-by-Step Solution:

1. **Understanding Instantaneous Speed**:
- Instantaneous speed is defined as the speed of an object at a specific moment in time. Here, it is given that \( v_0 \) is a positive constant. This means that the object is moving at a constant speed throughout its motion.

2. **Analyzing the Options**:
- We have four statements to evaluate:
1. Average velocity is not 0 at any time.
2. Average acceleration must always vanish.
3. Displacement in equal time intervals are equal.
4. Equal path lengths are traversed in equal time intervals.

3. **Evaluating Option 4**:
- Since the speed \( v_0 \) is constant, we can say that the distance covered in equal time intervals will also be equal. This is because speed is defined as distance divided by time. Thus, if time intervals are equal and speed is constant, the distances must also be equal.
- **Conclusion**: Option 4 is necessarily true.

4. **Evaluating Option 3**:
- Displacement is a vector quantity and depends on the direction of motion. While the distance covered in equal time intervals is constant, the direction may change, leading to different displacements. Therefore, we cannot conclude that the displacements in equal time intervals are equal.
- **Conclusion**: Option 3 is not necessarily true.

5. **Evaluating Option 2**:
- Average acceleration is defined as the change in velocity over time. Since we only know that speed is constant and not velocity, we cannot conclude that the average acceleration must vanish. For example, in uniform circular motion, the speed is constant, but the direction changes, resulting in a non-zero average acceleration.
- **Conclusion**: Option 2 is not necessarily true.

6. **Evaluating Option 1**:
- Average velocity is defined as the total displacement divided by the total time. Since the direction of motion can change while maintaining constant speed, it is possible for the average velocity to be zero over a complete cycle (for example, in circular motion). Thus, we cannot assert that the average velocity is not zero at any time.
- **Conclusion**: Option 1 is not necessarily true.

### Final Answer:
Among the four options, only **Option 4** ("Equal path lengths are traversed in equal time intervals") is necessarily true.