Which of the following statements is /are correct ?
I.Average speed over a finite interval of times is greater or equal to the magnitude of the average velocity.
The instantaneous speed at an instant is equal to the magnitude of the instantaneous velocity at that instant.

Options

A. Only I
B. Only II
C. Both I and II
D. None is correct

Answer

To determine which of the given statements are correct, we will analyze each statement step by step.

### Step-by-Step Solution:

**Step 1: Analyze Statement I**
- Statement I: "Average speed over a finite interval of time is greater than or equal to 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. Mathematically, it is given by:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]

- **Definition of Average Velocity**: Average velocity is defined as the total displacement (change in position) divided by the total time taken. Mathematically, it is given by:
\[ \text{Average Velocity} = \frac{\text{Total Displacement}}{\text{Total Time}} \]

- **Comparison**: The key difference is that average speed considers the total distance traveled, while average velocity considers only the displacement between the initial and final positions. Since displacement is the shortest path between two points, the total distance traveled will always be greater than or equal to the displacement. Thus:
\[ \text{Average Speed} \geq |\text{Average Velocity}| \]
- **Conclusion for Statement I**: This statement is correct.

**Step 2: Analyze Statement II**
- Statement II: "The instantaneous speed at an instant is equal to the magnitude of the instantaneous velocity at that instant."
- **Definition of Instantaneous Speed**: Instantaneous speed is the speed of an object at a specific moment in time, calculated as the limit of the average speed as the time interval approaches zero:
\[ \text{Instantaneous Speed} = \lim_{\Delta t \to 0} \frac{\Delta s}{\Delta t} \]

- **Definition of Instantaneous Velocity**: Instantaneous velocity is the velocity of an object at a specific moment in time, which includes both speed and direction. It is also calculated as:
\[ \text{Instantaneous Velocity} = \lim_{\Delta t \to 0} \frac{\Delta s}{\Delta t} \]

- **Magnitude Comparison**: The magnitude of instantaneous velocity is simply the speed at that instant. Therefore:
\[ \text{Instantaneous Speed} = |\text{Instantaneous Velocity}| \]
- **Conclusion for Statement II**: This statement is also correct.

### Final Conclusion:
Both statements are correct.