https://file.aiquickdraw.com/mathbot/file/1725770403305cmlcqj94.jpg

It looks like you've uploaded an image related to some problems in physics or calculus that involve motion, graphs, and possibly velocity or acceleration. Let me carefully analyze the image and extract the content.

### Problem 1:
The graph appears to represent the velocity (v) of a particle as a function of time (t), with units in meters per second (m/s) on the y-axis and seconds (s) on the x-axis. There are intervals where the velocity is both positive and negative, meaning the particle changes direction.

The sub-questions for this problem are:
1. **Where is the particle's velocity zero?** 
    - From the graph, this occurs at specific times where the velocity crosses the x-axis (t-axis), such as \( t = 3 \, \text{seconds}, t = 6 \, \text{seconds}, \) and \( t = 9 \, \text{seconds} \).

2. **When is the particle moving forward and when is it moving backward?**
    - Forward motion occurs when velocity is positive.
    - Backward motion occurs when velocity is negative.

3. **What is the particle’s acceleration between the points?** (Likely asked for each interval)
    - The acceleration would be the slope of the velocity-time graph. So, for each interval, you'd need to calculate the change in velocity divided by the change in time (\( a = \frac{\Delta v}{\Delta t} \)).

### Problem 2:
The next problem seems to focus on analyzing or interpreting specific events (perhaps finding displacement or specific times when velocity is zero).

### Completion of the table:
There’s a table in the middle section with time points and velocities where you are likely expected to calculate and complete certain values based on information provided by the graph.

Let me know if you'd like a more detailed breakdown for any specific part of the image, and I can work through each calculation or explanation step-by-step!

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**Relative Questions:**
1. How do you find the acceleration from a velocity-time graph?
2. What does the area under a velocity-time graph represent?
3. Why does velocity change from positive to negative in such problems?
4. How can you interpret motion graphs in terms of displacement?
5. What would happen if the graph was shifted or had different slopes?

**Tip:** When interpreting graphs, always remember that the slope gives you acceleration (in velocity-time graphs), and the area gives you displacement.

The original question submitted by the user as text.

This problem involves interpreting a velocity-time graph to determine when velocity is zero, the direction of motion, and acceleration. Learn how to calculate these values and analyze the graph for a particle's motion. Suitable for Grades 10-12.

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