This problem requires interpreting the given graph of position $x$ versus time $t$ and constructing a velocity ($|v|$) versus time ($t$) graph based on the motion described.

Solution Steps:

1. Analyze the Position-Time Graph:

   • Segment $t_1 \to t_2$: The slope is negative, indicating a constant velocity in the negative direction.
   • Segment $t_2 \to t_3$: The slope becomes positive, indicating a constant velocity in the positive direction.
   • Between Segments $t_1, t_2$ and $t_3$: The graph is linear, so the object has constant velocities during these intervals.

2. Relate Slope to Velocity:

   • The slope of the $x(t)$ graph equals the velocity $v(t)$.
   • The velocity changes sign at $t_2$, indicating a change in direction.

3. Draw the Velocity Magnitude Graph ($|v|$):

   • Use the slopes of each segment to determine the velocity's absolute value.
   • Plot the absolute value of velocity as a function of time.

Create the $|v|$-Time Graph:

Let me plot the graph to visualize this.import matplotlib.pyplot as plt

Define time intervals and corresponding velocity magnitudes

Time points: t1, t2, t3

times = [0, 2, 4, 6] # Assume t1=2, t2=4, t3=6 for simplicity velocities = [0, 2, 2, 0] # Corresponding absolute velocities based on the slopes

Plot the velocity-time graph

plt.figure(figsize=(8, 5)) plt.plot(times, velocities, marker='o', linestyle='-', label="|v| vs Time") plt.title("Velocity Magnitude |v| vs Time") plt.xlabel("Time (t)") plt.ylabel("Velocity Magnitude (|v|)") plt.grid(True) plt.legend() plt.show() <<ImageDisplayed>>successHere is the velocity magnitude ($|v|$) versus time ($t$) graph derived from the position-time graph. The graph shows:

• A steady increase in $|v|$ from $t=0$ to $t_2$.
• Constant $|v|$ from $t_2$ to $t_3$.
• A steady decrease in $|v|$ after $t_3$.

Let me know if you need further explanation or if you have additional questions.

Related Questions:

1. How is the velocity graph different from the speed graph?
2. What does the change in slope at $t_2$ imply about the motion?
3. How can acceleration be determined from the position-time graph?
4. What real-world scenario could represent this type of motion?
5. How does the motion change if the position-time graph is quadratic?

Tip:

Always derive velocity by calculating the slope of the position-time graph, and derive acceleration from the slope of the velocity-time graph.