Let `bar(v)(t)` be the velocity of a particle al lime t. Then :

Options

A. `|dbar(v)(t)//dt|and d|bar(v)(t)|//dt|"are always equal"`
B. `|dbar(v)(t)//dt|and d|bar(v)(t)|//dt|",may always equal"`
C. `|dbar(v)(t)//dt |"can be zero while"and d|bar(v)(t)|//dt|",is not zero"`
D. `d|bar(v)(t)|//dt|ne0"implies " |d bar(v)(t)//dt|ne0 `

Answer

To solve the problem regarding the relationship between the velocity of a particle and its derivatives, we need to analyze the options given and understand the mathematical implications of taking the modulus and the derivative.

### Step-by-Step Solution:

1. **Understanding Velocity and Derivatives**:
- Let \( v(t) \) be the velocity of a particle at time \( t \).
- The derivative of velocity with respect to time is given by \( \frac{dv}{dt} \), which represents acceleration.

2. **Modulus of Velocity**:
- The modulus of the velocity is denoted as \( |v(t)| \). This means that all negative values of \( v(t) \) are turned into positive values.

3. **Taking Derivatives**:
- The derivative of the modulus of velocity with respect to time is \( \frac{d|v|}{dt} \).
- The key point here is that \( |v(t)| \) changes the sign of negative velocities, which affects the slope when we take the derivative.

4. **Comparing the Two Derivatives**:
- We need to compare \( \left| \frac{dv}{dt} \right| \) and \( \frac{d|v|}{dt} \).
- If \( v(t) \) is negative, \( \frac{dv}{dt} \) will also be negative, making \( \left| \frac{dv}{dt} \right| \) positive.
- However, \( \frac{d|v|}{dt} \) will be positive since we are taking the derivative of a positive value.

5. **Analyzing the Options**:
- **Option A**: States that \( \left| \frac{dv}{dt} \right| = \frac{d|v|}{dt} \) is always true. This is incorrect because they can have different signs.
- **Option B**: Similar reasoning applies; it is incorrect.
- **Option C**: States that \( \frac{dv}{dt} \) can be 0 while \( \frac{d|v|}{dt} \) is not 0. This is also incorrect because if \( \frac{dv}{dt} = 0 \), then \( v(t) \) is constant, and thus \( |v(t)| \) would also not change.
- **Option D**: This option is correct as it states that \( \left| \frac{dv}{dt} \right| \) and \( \frac{d|v|}{dt} \) can be equal under certain conditions.

### Conclusion:
From the analysis, we conclude that the correct relationship is that the modulus of the derivative of velocity can differ from the derivative of the modulus of velocity depending on the sign of the velocity.