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 is equal time intervals are equal.
D. Equal path lengths are traversed in equal intervals.

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 evaluate each statement one by one.

### Step 1: Understanding Instantaneous Speed
Instantaneous speed is defined as the magnitude of the instantaneous velocity at a given point in time. Since \( v_0 \) is a positive constant, this means that the object is moving at a constant speed throughout its motion.

### Step 2: Evaluating Each Statement

1. **Statement 1: Average velocity is not zero at any time.**
- Average velocity is defined as the total displacement divided by the total time taken. Since we only know that speed is constant, we cannot determine the direction of motion. If the object moves in a closed path (like a circle), the average velocity can indeed be zero. Therefore, this statement is **not necessarily true**.

2. **Statement 2: Average acceleration must always be a vector quantity.**
- Average acceleration is defined as the change in velocity divided by the time taken. It is always a vector quantity, regardless of the speed being constant. However, we cannot conclude anything about its magnitude or direction without additional information. Thus, while this statement is true in terms of definition, it does not provide any new information about the motion. Therefore, this statement is **not necessarily true** in the context of the problem.

3. **Statement 3: Displacement in equal time intervals are equal.**
- Since the speed is constant, the distance covered in equal time intervals will be the same. However, displacement is a vector quantity and depends on the direction of motion. If the motion is not linear, the displacement may not be equal even if the distance is. Therefore, this statement is **not necessarily true**.

4. **Statement 4: Equal path lengths are traversed in equal intervals.**
- Since the instantaneous speed \( v_0 \) is constant, the distance (or path length) covered in equal time intervals will be the same. This is a direct consequence of the definition of speed (distance/time). Therefore, this statement is **necessarily true**.

### Summary of Results
- **Statement 1:** Not necessarily true.
- **Statement 2:** Not necessarily true.
- **Statement 3:** Not necessarily true.
- **Statement 4:** Necessarily true.

### Final Answer
The only statement that is necessarily true is **Statement 4: Equal path lengths are traversed in equal intervals.**