Conversion of displacement-time graphs to velocity-time graphs for different motion scenarios.

Let's solve this step by step.

We are given Displacement vs. Time (d vs. t) graphs and need to convert them into Velocity vs. Time (v vs. t) graphs.

Key Concept:

- Velocity is the slope of the displacement-time graph.
- If the line is straight, the slope is constant → constant velocity.
- If the line is horizontal (flat), slope = 0 → velocity = 0.
- Positive slope → positive velocity.
- Negative slope → negative velocity.
- Steeper slope → higher speed.

We’ll analyze each of the four displacement graphs and draw the corresponding velocity graph below it.

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🔹 Graph 1 (Top Left)

Displacement vs. Time:
- Starts at (0,0) and goes to (7,7) in a straight line.
- Constant positive slope.

Slope = Δd / Δt = (7 - 0)/(7 - 0) = 1 m/s

✔ So, velocity is constant at +1 m/s for all time from 0 to 7 seconds.

Velocity vs. Time Graph:
- A horizontal line at v = 1 m/s from t = 0 to t = 7.

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🔹 Graph 2 (Top Middle-Left)

Displacement vs. Time:
- From t = 0 to t = 2: goes from (0,0) to (2,6) → steep upward line.
- From t = 2 to t = 7: flat line at d = 6.

Segment 1 (0 to 2):
- Slope = (6 - 0)/(2 - 0) = 3 m/s → velocity = +3 m/s

Segment 2 (2 to 7):
- Flat line → slope = 0 → velocity = 0

Velocity vs. Time Graph:
- From t = 0 to t = 2: horizontal line at v = 3 m/s
- From t = 2 to t = 7: horizontal line at v = 0

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🔹 Graph 3 (Top Middle-Right)

Displacement vs. Time:
- From t = 0 to t = 2: from (0,6) to (2,2) → downward slope
- From t = 2 to t = 4: from (2,2) to (4,4) → upward slope
- From t = 4 to t = 7: flat at d = 4

Segment 1 (0 to 2):
- Slope = (2 - 6)/(2 - 0) = (-4)/2 = -2 m/s

Segment 2 (2 to 4):
- Slope = (4 - 2)/(4 - 2) = 2/2 = +1 m/s

Segment 3 (4 to 7):
- Flat → slope = 0 → velocity = 0

Velocity vs. Time Graph:
- t = 0 to 2: v = -2 m/s
- t = 2 to 4: v = +1 m/s
- t = 4 to 7: v = 0

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🔹 Graph 4 (Top Right)

Displacement vs. Time:
- From t = 0 to t = 4: from (0,5) to (4,7) → upward slope
- From t = 4 to t = 6: from (4,7) to (6,1) → steep downward
- From t = 6 to t = 7: flat at d = 1

Segment 1 (0 to 4):
- Slope = (7 - 5)/(4 - 0) = 2/4 = +0.5 m/s

Segment 2 (4 to 6):
- Slope = (1 - 7)/(6 - 4) = (-6)/2 = -3 m/s

Segment 3 (6 to 7):
- Flat → slope = 0 → velocity = 0

Velocity vs. Time Graph:
- t = 0 to 4: v = +0.5 m/s
- t = 4 to 6: v = -3 m/s
- t = 6 to 7: v = 0

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✔ Final Answer: Draw these on the green grids

Here’s how to fill in each V vs. T graph:

#### 🟩 Graph 1 (Bottom Left):
- Draw a horizontal line at v = 1 from t = 0 to 7.

#### 🟩 Graph 2 (Bottom Middle-Left):
- From t = 0 to 2: v = 3
- From t = 2 to 7: v = 0
- Draw a horizontal line at 3, then drop to 0 at t=2.

#### 🟩 Graph 3 (Bottom Middle-Right):
- t = 0 to 2: v = -2
- t = 2 to 4: v = +1
- t = 4 to 7: v = 0
- Draw:
- Horizontal line at -2 (0–2)
- Then jump up to +1 (2–4)
- Then drop to 0 (4–7)

#### 🟩 Graph 4 (Bottom Right):
- t = 0 to 4: v = +0.5
- t = 4 to 6: v = -3
- t = 6 to 7: v = 0
- Draw:
- Line at +0.5 (0–4)
- Drop to -3 (4–6)
- Then to 0 (6–7)

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Summary Table:

| Graph | Velocity Intervals |
|-------|---------------------|
| 1 | v = 1 m/s (0–7) |
| 2 | v = 3 m/s (0–2), v = 0 (2–7) |
| 3 | v = -2 m/s (0–2), v = +1 m/s (2–4), v = 0 (4–7) |
| 4 | v = +0.5 m/s (0–4), v = -3 m/s (4–6), v = 0 (6–7) |

Now you can draw these on the provided green grids accordingly.

Let me know if you'd like me to generate an image or sketch of the correct velocity graphs!