The instantaneous velocity of a particle is equal to time derivative of its position vector and the instantaneous acceleration is equal to time derivatives of its velocity vector. Therefore `:`

Options

A. the instantaneous velocity depends on the instantaneous position vecoty
B. instantaneous acceleration is independent of instantaneous position vector and instantaneous velocity
C. instantaneous acceleration is independent of instantaneous position vector but depends on the instantaneous velocity
D. instantaneous acceleration depends both on the instataneous position vector and the instantaneous velocity.

Answer

To solve the problem, we need to analyze the statements regarding instantaneous velocity and instantaneous acceleration in relation to the position vector and velocity vector of a particle.

### Step-by-Step Solution:

1. **Understanding Instantaneous Velocity**:
- The instantaneous velocity \( \mathbf{v} \) of a particle is defined as the time derivative of its position vector \( \mathbf{r} \):
\[ \mathbf{v} = \frac{d\mathbf{r}}{dt} \]
- This means that velocity is related to how the position vector changes with time.

2. **Understanding Instantaneous Acceleration**:
- The instantaneous acceleration \( \mathbf{a} \) of a particle is defined as the time derivative of its velocity vector \( \mathbf{v} \):
\[ \mathbf{a} = \frac{d\mathbf{v}}{dt} \]
- This indicates that acceleration is related to how the velocity vector changes with time.

3. **Analyzing the Options**:
- **Option 1**: "Instantaneous velocity depends on the instantaneous position vector."
- This statement is incorrect because instantaneous velocity is derived from the position vector but does not depend directly on its value at any instant. It depends on how the position changes over time (the derivative).

- **Option 2**: "Instantaneous acceleration is independent of instantaneous position vector and instantaneous velocity."
- This statement is also incorrect. Acceleration depends on the instantaneous velocity (as shown in the definition of acceleration) and is not independent of it.

- **Option 3**: "Instantaneous acceleration is independent of instantaneous position vector but depends on instantaneous velocity."
- This statement is correct. Acceleration is related to how velocity changes over time and does not depend on the position vector directly.

- **Option 4**: "Instantaneous acceleration depends on both the instantaneous position vector and instantaneous velocity."
- This statement is incorrect because, as established, acceleration does not depend on the instantaneous position vector.

4. **Conclusion**:
- The only correct statement is **Option 3**: "Instantaneous acceleration is independent of instantaneous position vector but depends on instantaneous velocity."