(a) What does `|(dv)/(dt)|` and `(d|v|)/(dt)` represent? (b) Can these be equal?

Explanation

To solve the question, we will break it down into two parts: (a) understanding what \( \left| \frac{dv}{dt} \right| \) and \( \frac{d|v|}{dt} \) represent, and (b) determining if these two quantities can be equal.

### Step-by-Step Solution:

**(a)**

1. **Understanding \( \left| \frac{dv}{dt} \right| \)**:
- The expression \( \frac{dv}{dt} \) represents the acceleration of an object, which is the rate of change of velocity with respect to time.
- Taking the absolute value, \( \left| \frac{dv}{dt} \right| \), gives us the magnitude of the total acceleration. This indicates how quickly the velocity of the object is changing, irrespective of the direction of that change.

2. **Understanding \( \frac{d|v|}{dt} \)**:
- The term \( |v| \) represents the speed of the object, which is the magnitude of the velocity vector \( v \).
- Thus, \( \frac{d|v|}{dt} \) denotes the rate of change of speed with respect to time. This is also known as the tangential acceleration, which specifically refers to the component of acceleration that changes the speed of the object.

**(b)**

3. **Can these two quantities be equal?**:
- Yes, \( \left| \frac{dv}{dt} \right| \) can be equal to \( \frac{d|v|}{dt} \) under certain conditions.
- This equality holds true when the motion of the object is in a straight line (one-dimensional motion). In such cases, the direction of the velocity does not change, and thus the rate of change of velocity (total acceleration) is equal to the rate of change of speed (tangential acceleration).

### Final Answer:
(a) \( \left| \frac{dv}{dt} \right| \) represents the magnitude of total acceleration, while \( \frac{d|v|}{dt} \) represents the tangential acceleration (rate of change of speed).
(b) Yes, these two quantities can be equal when the body is in straight line motion (1D motion).