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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.

The position of a particle is given by x(t) = t^2 + 4t - 2 m. What is the average velocity during the time interval from t1 = 1 s to t2 = 2 s?

This problem explores the concept of average velocity using the position function x(t) = t^2 + 4t - 2. Learn how to find the average velocity over a specific time interval from t1 = 1 s to t2 = 2 s. Suitable for students in Grades 10-12.

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