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The graph provided shows the motion of a particle along the x-axis as a function of time, with the following tasks:

### (a) Find the average velocity of the particle during the time interval from \( t = 1.50 \, \text{s} \) to \( t = 4.00 \, \text{s} \).

The formula for average velocity \( v_{avg} \) is given by:
\[
v_{avg} = \frac{x(t_2) - x(t_1)}{t_2 - t_1}
\]
We need the values of \( x(t_1) \) at \( t_1 = 1.50 \, \text{s} \) and \( x(t_2) \) at \( t_2 = 4.00 \, \text{s} \) from the graph. After reading from the graph:
- \( x(1.5) \) appears to be approximately 9 units.
- \( x(4.0) \) appears to be approximately 2 units.

Using these values:
\[
v_{avg} = \frac{2 - 9}{4.00 - 1.50} = \frac{-7}{2.5} = -2.8 \, \text{units/s}
\]

### (b) Find the instantaneous velocity at \( t = 2.00 \, \text{s} \).

The instantaneous velocity at any point is the slope of the tangent to the curve at that point. Based on the graph, we can estimate the slope of the tangent line at \( t = 2.00 \, \text{s} \).

The green line appears to be a tangent at \( t = 2.00 \, \text{s} \), which passes through \( t = 1.00 \, \text{s} \) and \( t = 3.00 \, \text{s} \). From the graph:
- \( x(1.00) \approx 10 \) units
- \( x(3.00) \approx 3 \) units.

The slope (instantaneous velocity) is:
\[
v = \frac{x(3.00) - x(1.00)}{3.00 - 1.00} = \frac{3 - 10}{3.00 - 1.00} = \frac{-7}{2} = -3.5 \, \text{units/s}
\]

### (c) At what time(s) is the velocity of the particle zero?

The velocity is zero when the slope of the tangent line is zero, which occurs at the turning point of the curve (the minimum point on the graph). This happens at approximately \( t = 4.5 \, \text{s} \).

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Would you like more details or have any questions?

Here are 5 related questions to consider:
1. How do you calculate instantaneous velocity if a tangent line is not provided?
2. Can average velocity be zero even if the particle moves?
3. How does negative velocity relate to the motion of the particle?
4. What is the difference between velocity and speed in physics?
5. How do we define acceleration from an x-t graph?

**Tip**: Instantaneous velocity gives the exact speed and direction at a specific moment, while average velocity considers the entire interval.

The graph x-t represents the motion of a particle along the x-axis over time with three sub-tasks:
(a) Find the average velocity of the particle during the time interval from t = 1.50 s to t = 4.00 s.
(b) Calculate the instantaneous velocity at t = 2.00 s using the slope of the tangent at that point.
(c) Identify the time when the velocity of the particle is zero.

This problem demonstrates how to calculate both the average and instantaneous velocity of a particle moving along the x-axis using a graphical approach. It also includes finding the point where velocity equals zero. Suitable for Grades 10-12.

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