Under what condition the magnitude of average velocity is equal to the average speed ?

Explanation

To determine the condition under which the magnitude of average velocity is equal to the average speed, we can follow these steps:

### Step-by-Step Solution:

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

\[ \text{Average Velocity} (v_{avg}) = \frac{\text{Total Displacement}}{\text{Total Time}} \]
\[ \text{Average Speed} (s_{avg}) = \frac{\text{Total Distance}}{\text{Total Time}} \]

2. **Setting the Condition for Equality**:
- For the magnitudes of average velocity and average speed to be equal, we can set up the following equation:

\[ |v_{avg}| = s_{avg} \]

This implies:
\[ \frac{\text{Total Displacement}}{\text{Total Time}} = \frac{\text{Total Distance}}{\text{Total Time}} \]

3. **Simplifying the Equation**:
- Since the total time is the same for both average velocity and average speed, we can cancel it out from both sides of the equation:

\[ \text{Total Displacement} = \text{Total Distance} \]

4. **Understanding Displacement and Distance**:
- Displacement is defined as the shortest distance between the initial and final positions of an object, while distance is the total path length traveled by the object.
- For the total displacement to equal the total distance, the object must not change direction.

5. **Conclusion**:
- The only scenario where the total displacement equals the total distance is when the object moves in a straight line without changing its direction. Therefore, the condition under which the magnitude of average velocity is equal to the average speed is:

**The object must move in a straight line.**