To determine the correct formula for average velocity, we need to understand the definition and calculation of average velocity. Average velocity is defined as the total displacement divided by the total time taken.
Let's denote:
- [tex]\( s(a) \)[/tex] as the initial position at time [tex]\( a \)[/tex]
- [tex]\( s(b) \)[/tex] as the final position at time [tex]\( b \)[/tex]
The displacement is the change in position, which can be calculated as [tex]\( s(b) - s(a) \)[/tex]. The total time taken is the difference in time, which is [tex]\( b - a \)[/tex].
Thus, the formula for average velocity [tex]\( V_{av} \)[/tex] is:
[tex]\[ V_{av} = \frac{\text{Displacement}}{\text{Time}} = \frac{s(b) - s(a)}{b - a} \][/tex]
Now, let's match this formula with the given multiple choice options:
A. [tex]\( V_{av} = \frac{s(a) + s(b)}{a - b} \)[/tex]
This formula does not correctly represent the average velocity because it incorrectly sums the positions and divides by the difference in time.
B. [tex]\( V_{av} = \frac{s(b) - s(a)}{b - a} \)[/tex]
This formula correctly represents the average velocity as it uses the displacement [tex]\( s(b) - s(a) \)[/tex] and divides by the time difference [tex]\( b - a \)[/tex].
C. [tex]\( V_{av} = \frac{s(b) + s(a)}{b - a} \)[/tex]
This formula incorrectly sums the positions instead of finding the difference in positions.
D. [tex]\( V_{av} = \frac{s(a) - s(b)}{b - a} \)[/tex]
This formula inverts the displacement calculation by subtracting [tex]\( s(b) \)[/tex] from [tex]\( s(a) \)[/tex], which is incorrect.
Comparing all the options, the correct formula for average velocity is:
[tex]\[ \text{Option } \text{B}. \][/tex]
Therefore, the correct answer is option B:
[tex]\[ V_{av} = \frac{s(b) - s(a)}{b - a} \][/tex]
Hence, the correct option is 2.
