https://file.aiquickdraw.com/mathbot/file/1725556183272sdyogz9z.jpg

It looks like you've uploaded an image containing a graph of the position of a particle over time, along with some related questions. Let's break it down and solve them step by step.

1. **Understanding the Graph:**
   - The graph represents the position \( x(t) \) of a particle as a function of time \( t \).
   - The horizontal axis is time \( t \) (in seconds), and the vertical axis is position \( x \) (in meters).

2. **First Question:**
   - "What is the particle's average velocity between times \( t = 0 \, \text{s} \) and \( t = 8 \, \text{s}? \)"

The average velocity \( v_{\text{avg}} \) is defined as:

\[
v_{\text{avg}} = \frac{\Delta x}{\Delta t} = \frac{x(t_{\text{final}}) - x(t_{\text{initial}})}{t_{\text{final}} - t_{\text{initial}}}
\]

Let's calculate the change in position between \( t = 0 \, \text{s} \) and \( t = 8 \, \text{s} \) based on the graph and then compute the average velocity.

3. **Second Question:**
   - "What is the average speed of the particle between times \( t = 6 \, \text{s} \) and \( t = 7 \, \text{s}? \)"

The average speed is the total distance traveled (regardless of direction) divided by the total time.

4. **Third Question:**
   - "What is the average acceleration of the particle in the time interval between \( t = 1.5 \, \text{s} \) and \( t = 4 \, \text{s}? \)"

Average acceleration \( a_{\text{avg}} \) is defined as:

\[
a_{\text{avg}} = \frac{\Delta v}{\Delta t}
\]

We will analyze the velocity changes between \( t = 1.5 \, \text{s} \) and \( t = 4 \, \text{s} \).

I'll extract the values from the graph and perform the calculations next.

The position of a particle as a function of time is shown in the figure. What is the particle's average velocity between times t = 0 and t = 8 s?

This page explains how to calculate the average velocity and acceleration of a particle based on its position-time graph. It covers key kinematics concepts suitable for students in Grades 9-12, including how to use kinematic formulas to find velocity and acceleration over specific intervals.

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