Assertion: Average speed of a particle in a given time interval is never less than the magnitude of the average velocity.
Reason: The magnitude of the velocity (instantaneous velocity) of a particle is equal to its speed.

Options

A. a.If both assertion `&` Reason are True `&` the Reason is a corrrect explanation of the Asserion.
B. b.If both Assertion `&` Reason are True but Reason is not correct explanation of the Assertion.
C. c.If Assertion is Trie but the Reason is False.
D. d.If both Assertion `&` Reason are false

Answer

To solve the given assertion and reason question, we will analyze both statements step by step.

### Step 1: Understanding the Assertion
**Assertion:** The average speed of a particle in a given time interval is never less than the magnitude of the average velocity.

- **Definition of Average Speed:** Average speed is defined as the total distance traveled divided by the total time taken.
- **Definition of Average Velocity:** Average velocity is defined as the total displacement divided by the total time taken.

Since distance is always greater than or equal to displacement (because distance accounts for the actual path taken, while displacement is the straight line from the initial to the final position), we can express this mathematically:

\[ \text{Distance} \geq \text{Displacement} \]

### Step 2: Dividing by Time
Now, if we divide both sides of the above inequality by the total time taken (which is always positive), we have:

\[ \frac{\text{Distance}}{\text{Time}} \geq \frac{\text{Displacement}}{\text{Time}} \]

This simplifies to:

\[ \text{Average Speed} \geq \text{Average Velocity} \]

Thus, the assertion is true.

### Step 3: Understanding the Reason
**Reason:** The magnitude of the velocity (instantaneous velocity) of a particle is equal to its speed.

- **Definition of Instantaneous Velocity:** Instantaneous velocity is defined as the limit of the average velocity as the time interval approaches zero. Mathematically, it is expressed as:

\[ v = \frac{dr}{dt} \]

Where \( dr \) is the change in position and \( dt \) is the change in time.

- **Magnitude of Velocity:** The magnitude of instantaneous velocity is indeed equal to speed, which is a scalar quantity representing how fast an object is moving regardless of its direction.

Thus, the reason is also true.

### Step 4: Conclusion
Both the assertion and the reason are true. However, the reason does not provide a correct explanation for the assertion. The assertion is about average speed and average velocity, while the reason discusses instantaneous velocity.

### Final Answer
- **Assertion:** True
- **Reason:** True, but does not explain the assertion.

The correct option is B: Both assertion and reason are true, but the reason is not the correct explanation of the assertion.

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