Assertion: Always `|(dvecv)/(dt)|=d/(dt)|vecv|` , where `vecv` has its usual meaning.
(Reason): Acceleration is rate of change of velocity.

Options

A. Both (Assertion) and (Reason) are true and (Reason) is the correct explanation of (Assertion)
B. Both (Assertion) and (Reason) are true and (Reason) is not the correct explanation of (Assertion)
C. (Assertion) is true but (Reason) is false
D. (Assertion) is false but (Reason) is true

Answer

To solve the question, we need to analyze the assertion and the reason provided:

**Assertion:** \( \left| \frac{d\vec{v}}{dt} \right| = \frac{d}{dt} |\vec{v}| \)

**Reason:** Acceleration is the rate of change of velocity.

### Step-by-Step Solution:

1. **Understand the Terms:**
- Let \( \vec{v} \) be the velocity vector.
- The magnitude of the velocity vector is denoted as \( |\vec{v}| \).

2. **Acceleration Definition:**
- Acceleration \( \vec{a} \) is defined as the rate of change of velocity:
\[ \vec{a} = \frac{d\vec{v}}{dt} \]

3. **Magnitude of Velocity:**
- The magnitude of the velocity vector \( |\vec{v}| \) is given by:
\[ |\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2} \]
- If \( \vec{v} \) is expressed in terms of its components, we can differentiate \( |\vec{v}| \) with respect to time.

4. **Differentiating the Magnitude:**
- Using the chain rule, we differentiate the magnitude of the velocity:
\[ \frac{d}{dt} |\vec{v}| = \frac{d}{dt} \left( \sqrt{v_x^2 + v_y^2 + v_z^2} \right) \]
- Applying the chain rule:
\[ \frac{d}{dt} |\vec{v}| = \frac{1}{2\sqrt{v_x^2 + v_y^2 + v_z^2}} \cdot \left( 2v_x \frac{dv_x}{dt} + 2v_y \frac{dv_y}{dt} + 2v_z \frac{dv_z}{dt} \right) \]
- This simplifies to:
\[ \frac{d}{dt} |\vec{v}| = \frac{1}{|\vec{v}|} \left( v_x \frac{dv_x}{dt} + v_y \frac{dv_y}{dt} + v_z \frac{dv_z}{dt} \right) \]

5. **Comparing Both Sides:**
- The left-hand side \( \left| \frac{d\vec{v}}{dt} \right| \) represents the magnitude of the acceleration vector.
- The right-hand side \( \frac{d}{dt} |\vec{v}| \) does not always equal \( \left| \frac{d\vec{v}}{dt} \right| \) unless the direction of the velocity vector remains constant.

6. **Conclusion:**
- The assertion \( \left| \frac{d\vec{v}}{dt} \right| = \frac{d}{dt} |\vec{v}| \) is not always true. It holds true only under specific conditions (e.g., when the direction of the velocity does not change).
- Therefore, the assertion is **false** while the reason is **true**.

### Final Answer:
- Assertion: False
- Reason: True