https://file.aiquickdraw.com/mathbot/file/1726795713426hiwgz5tw.jpg

Let's break down the problem based on the velocity-time graph and answer each question:

**(a) When does the particle move forward? move backward? speed up? slow down?**

- **Move forward**: The particle moves forward when its velocity is positive. From the graph, this happens from \( t = 0 \) to \( t = 3 \) and from \( t = 5 \) to \( t = 8 \) because the velocity is above the time-axis.
  
- **Move backward**: The particle moves backward when its velocity is negative. From the graph, this occurs from \( t = 3 \) to \( t = 5 \) because the velocity is below the time-axis.
  
- **Speed up**: The particle speeds up when velocity and acceleration have the same sign (either both positive or both negative). This occurs when the slope of the graph matches the sign of the velocity.
  
- **Slow down**: The particle slows down when velocity and acceleration have opposite signs. For example, if the particle has a positive velocity and a negative slope (deceleration), it slows down.

**(b) When is the particle's acceleration positive? negative? zero?**

- **Positive acceleration**: Acceleration is positive when the velocity is increasing, which corresponds to intervals where the slope of the velocity graph is positive. This happens from \( t = 0 \) to \( t = 1 \), \( t = 3 \) to \( t = 4 \), and \( t = 6 \) to \( t = 7 \).
  
- **Negative acceleration**: Acceleration is negative when the velocity is decreasing, corresponding to intervals where the slope of the velocity graph is negative. This happens from \( t = 1 \) to \( t = 3 \), \( t = 4 \) to \( t = 6 \), and \( t = 7 \) to \( t = 8 \).
  
- **Zero acceleration**: Acceleration is zero when the velocity is constant. This happens from \( t = 8 \) to \( t = 10 \), where the velocity graph is flat.

**(c) When does the particle move at its greatest speed?**

The greatest speed occurs at the point where the velocity is farthest from zero, regardless of whether it's positive or negative. In the graph, the greatest speed is \( 5 \, \text{units/sec} \) and this occurs at \( t = 1 \) and \( t = 7 \).

**(d) When does the particle stand still for more than an instant?**

The particle stands still when the velocity is zero. This occurs from \( t = 8 \) to \( t = 10 \), as the velocity is zero over that entire interval.

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Do you need further clarification or a more detailed explanation?

Here are some relative questions to explore further:
1. What happens if the particle has zero velocity but nonzero acceleration?
2. How can we determine displacement from this velocity-time graph?
3. What is the total distance traveled by the particle based on the graph?
4. How would a position-time graph look for this particle's motion?
5. What would happen if the particle started with an initial position of zero?

**Tip**: The area under a velocity-time graph between two points gives the displacement of the particle over that interval.

The accompanying figure shows the velocity v = f(t) of a particle moving on a horizontal coordinate line. (a) When does the particle move forward? move backward? speed up? slow down? (b) When is the particle's acceleration positive? negative? zero? (c) When does the particle move at its greatest speed? (d) When does the particle stand still for more than an instant?

Analyze the velocity-time graph of a particle to understand when it moves forward, backward, speeds up, slows down, and stands still. Learn about velocity, acceleration, and the conditions under which the particle changes speed.

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