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 problem, we need to analyze the implications of having a constant instantaneous speed \( v_0 \) in two-dimensional motion. Let's break down the options provided in the question step by step.

### Step 1: Understand Instantaneous Speed
Instantaneous speed is defined as the magnitude of the instantaneous velocity at a specific point in time. Since \( v_0 \) is a positive constant, it means that the object is moving at a constant speed at all times.

### Step 2: Analyze the Options
We need to evaluate the truth of each statement given the condition of constant instantaneous speed.

#### 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 we only know that the instantaneous speed is constant, we cannot conclude anything about the direction of motion. If the object moves in a closed path, the displacement could be zero, making the average velocity zero. Therefore, this statement is **not necessarily true**.

#### Option 2: Average acceleration must always vanish.
- Average acceleration is defined as the change in velocity divided by the time interval. Since the speed is constant, the velocity could still change direction (as in circular motion), meaning there can be a non-zero average acceleration. Thus, this statement is also **not necessarily true**.

#### Option 3: Equal path lengths are traversed in equal intervals of time.
- Since the instantaneous speed \( v_0 \) is constant, the distance traveled in equal time intervals will also be constant. Therefore, if we take equal time intervals, the path lengths (distances) covered will also be equal. This statement is **necessarily true**.

### Conclusion
From the analysis:
- Option 1: Not necessarily true.
- Option 2: Not necessarily true.
- Option 3: Necessarily true.

### Final Answer
The only statement that is necessarily true is that equal path lengths are traversed in equal intervals of time.

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