To find the average velocity of an object moving along a straight line over a time interval $[1,3]$, we use the formula for average velocity:

$\text{Average velocity} = \frac{s(b) - s(a)}{b - a}$

Where:

• $s(t)$ is the position function,
• $a = 1$ and $b = 3$ are the endpoints of the interval.

Given the position function $s(t) = 1 + 2t + t^2$, we need to compute $s(3)$ and $s(1)$.

1. $s(3) = 1 + 2(3) + (3)^2 = 1 + 6 + 9 = 16$
2. $s(1) = 1 + 2(1) + (1)^2 = 1 + 2 + 1 = 4$

Now, we can find the average velocity:

$\text{Average velocity} = \frac{s(3) - s(1)}{3 - 1} = \frac{16 - 4}{2} = \frac{12}{2} = 6$

Thus, the average velocity of the object over the interval $[1,3]$ is 6 units per second.

Would you like more details or further clarification? Here are some related questions to deepen your understanding:

1. How do we interpret average velocity in terms of displacement and time?
2. Can you derive the instantaneous velocity from the given position function?
3. What is the object's velocity at $t = 2$?
4. How would the average velocity change if the interval was $[2,4]$?
5. How does the position function's quadratic nature affect the motion of the object?

Tip: To find the instantaneous velocity of an object, take the derivative of the position function with respect to time.