Question Statement

Using the following data, draw a time-displacement graph for a moving object:

Use the graph to find:

• Average velocity for the first 4 s
• Average velocity for the next 4 s
• Average velocity for the last 6 s
• The total displacement

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Background and Explanation

Average velocity is defined as the slope of the displacement-time graph, calculated as the change in displacement divided by the time interval: $v_{\text{avg}}=\frac{\Delta s}{\Delta t}$. The total displacement represents the net change in position from the initial to the final time.

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Solution

Drawing the Displacement-Time Graph

Plot time on the horizontal axis and displacement on the vertical axis. Mark the given data points and connect them with straight line segments:

Average Velocity for First 4 s (t = 0 s to t = 4 s)

From the displacement-time graph, identify the coordinates:

• At $t=0$ s: $s=0$ m
• At $t=4$ s: $s=4$ m

Calculate the slope (average velocity):

$v_{\text{avg}}=\frac{\Delta s}{\Delta t}=\frac{4-0}{4-0}=\frac{4}{4}=1\text{ m/s}$

Average Velocity for Next 4 s (t = 4 s to t = 8 s)

Identify the coordinates:

• At $t=4$ s: $s=4$ m
• At $t=8$ s: $s=4$ m

Calculate the average velocity:

$v_{\text{avg}}=\frac{\Delta s}{\Delta t}=\frac{4-4}{8-4}=\frac{0}{4}=0\text{ m/s}$

Interpretation: The object is at rest during this interval as its position does not change.

Average Velocity for Last 6 s (t = 10 s to t = 16 s)

The last 6 seconds of the 16-second motion corresponds to the interval from $t=10$ s to $t=16$ s.

Identify the coordinates:

• At $t=10$ s: $s=6$ m
• At $t=16$ s: $s=0$ m

Calculate the average velocity:

$v_{\text{avg}}=\frac{\Delta s}{\Delta t}=\frac{0-6}{16-10}=\frac{-6}{6}=-1\text{ m/s}$

Interpretation: The negative sign indicates the object is moving in the opposite direction (returning toward the origin).

Total Displacement

Total displacement is the net change in position from the start to the end of the motion:

$\text{Total Displacement}=s_{\text{final}}-s_{\text{initial}}=0\text{ m}-0\text{ m}=0\text{ m}$

The object returns to its starting point, resulting in zero net displacement.

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Key Formulas or Methods Used

• Average Velocity: $v_{\text{avg}}=\frac{\Delta s}{\Delta t}=\frac{s_2-s_1}{t_2-t_1}$
• Slope of Displacement-Time Graph: Represents velocity (positive slope = forward motion, zero slope = at rest, negative slope = backward motion)
• Total Displacement: $\Delta s_{\text{total}}=s_{\text{final}}-s_{\text{initial}}$

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Summary of Steps

1. Plot the graph: Mark displacement values against corresponding time values and connect with line segments
2. First 4 s: Calculate slope between $(0,0)$ and $(4,4)$ to get $v=1$ m/s
3. Next 4 s: Calculate slope between $(4,4)$ and $(8,4)$ to get $v=0$ m/s (object stationary)
4. Last 6 s: Calculate slope between $(10,6)$ and $(16,0)$ to get $v=-1$ m/s (negative indicates reverse direction)
5. Total displacement: Subtract initial position from final position: $0-0=0$ m