Assertion`:-` If a particle is acted upon by an external force its momentum must change
Reason`:-``|(dvec(v))/(dt)|` is ALWAYS equal to `(d)/(dt)|vec(v)|`
Here `vec(v)` has its usual meaning.

Options

A. If both Assertion & Reason are True & the Reason is a correct explanation of the Assertion.
B. If both Assertion & Reason are True but Reason is not a correct explanation of the Assertiion.
C. If Assertion is True but the Reason is False.
D. If both Assertion & Reason are False

Explanation

To solve the problem, we need to analyze the assertion and reason provided in the question.

### Step 1: Analyze the Assertion
The assertion states that if a particle is acted upon by an external force, its momentum must change. According to Newton's second law, the force acting on an object is equal to the rate of change of momentum of that object. Mathematically, this can be expressed as:

\[ F = \frac{d\vec{p}}{dt} \]

where \( \vec{p} \) is the momentum of the particle. If there is a non-zero external force acting on the particle, then the rate of change of momentum (\( \frac{d\vec{p}}{dt} \)) will also be non-zero, indicating that the momentum of the particle must change. Therefore, the assertion is **true**.

### Step 2: Analyze the Reason
The reason states that \( \left|\frac{d\vec{v}}{dt}\right| \) is always equal to \( \frac{d}{dt}|\vec{v}| \). Here, \( \vec{v} \) represents the velocity vector of the particle.

To understand this, we need to differentiate between the magnitude of the velocity vector and the rate of change of the velocity vector. The term \( \left|\frac{d\vec{v}}{dt}\right| \) represents the magnitude of the acceleration vector, while \( \frac{d}{dt}|\vec{v}| \) represents the rate of change of the speed (the magnitude of the velocity).

These two quantities are not always equal because the acceleration can change the direction of the velocity without changing its magnitude (speed). For instance, in circular motion, the speed remains constant while the direction of the velocity changes. Therefore, the reason is **false**.

### Conclusion
Based on the analysis:
- The assertion is **true**.
- The reason is **false**.

Thus, the correct answer is that the assertion is true, and the reason is false.