To find the average velocity of a particle over a given time interval, we use the formula:

$\text{Average Velocity} = \frac{x(t_2) - x(t_1)}{t_2 - t_1}$

where $x(t)$ is the position function, $t_1 = 1$ s, and $t_2 = 2$ s.

Given the position function:

$x(t) = t^2 + 4t - 2$

Let's calculate $x(t_1)$ and $x(t_2)$.

1. Calculate $x(1)$:

$x(1) = (1)^2 + 4(1) - 2 = 1 + 4 - 2 = 3$

2. Calculate $x(2)$:

$x(2) = (2)^2 + 4(2) - 2 = 4 + 8 - 2 = 10$

3. Calculate the Average Velocity:

$\text{Average Velocity} = \frac{x(2) - x(1)}{2 - 1} = \frac{10 - 3}{1} = 7 \, \text{m/s}$

Answer:

The average velocity of the particle during the time interval from $t = 1$ s to $t = 2$ s is 7 m/s.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. What is the instantaneous velocity of the particle at $t = 1.5$ s?
2. How do we find the acceleration of the particle at any given time $t$?
3. What is the significance of average velocity versus instantaneous velocity in motion?
4. How would the average velocity change if the time interval were from $t = 0$ to $t = 3$ s?
5. How can the position function be derived if given the velocity function?

Tip: Average velocity can provide a general sense of motion over an interval, but instantaneous velocity gives a more precise understanding at specific moments.