Converting displacement vs. time graphs to velocity vs. time graphs in physics.

Great! Let's go through this step-by-step.

You're asked to convert Displacement vs. Time (d-t) graphs into Velocity vs. Time (v-t) graphs. The key idea is:

> Velocity is the slope of the displacement-time graph.

So, for each segment of a displacement vs. time graph:
- If the line is straight, the slope is constant → velocity is constant.
- If the line is horizontal (flat), the slope is zero → velocity = 0.
- If the line is upward sloping, velocity is positive.
- If the line is downward sloping, velocity is negative.
- The steeper the slope, the greater the magnitude of velocity.

We’ll analyze each of the four d-t graphs and match them with the correct v-t graph.

---

🔹 Graph 1 (Top-left)

Displacement vs. Time:
- Starts at (0,0), ends at (7,7)
- Straight line with a constant positive slope
- Slope = rise/run = (7 - 0)/(7 - 0) = 1 m/s
- So, velocity = +1 m/s throughout

✔ This matches the first v-t graph (bottom-left): a horizontal line at v = 1 m/s

✔️ Match: Graph 1 → Bottom-left V graph

---

🔹 Graph 2 (Top-middle-left)

Displacement vs. Time:
- From t=0 to t=2: goes from (0,0) to (2,6) → slope = 6/2 = 3 m/s
- From t=2 to t=7: flat line → slope = 0 → 0 m/s

So:
- 0–2 s: v = +3 m/s
- 2–7 s: v = 0 m/s

✔ This matches the second v-t graph (bottom-middle-left):
- Horizontal line at v = 3 m/s from t=0 to t=2
- Then drops to v = 0 m/s from t=2 to t=7

✔️ Match: Graph 2 → Second V graph

---

🔹 Graph 3 (Top-middle-right)

Displacement vs. Time:
- From t=0 to t=2: from (0,6) to (2,2) → slope = (2 - 6)/(2 - 0) = (-4)/2 = -2 m/s
- From t=2 to t=4: from (2,2) to (4,4) → slope = (4 - 2)/(4 - 2) = 2/2 = +1 m/s
- From t=4 to t=7: flat line → slope = 0 → 0 m/s

So:
- 0–2 s: v = -2 m/s
- 2–4 s: v = +1 m/s
- 4–7 s: v = 0 m/s

✔ This matches the third v-t graph (bottom-middle-right):
- v = -2 m/s (from t=0 to t=2)
- Then jumps to +1 m/s (t=2 to t=4)
- Then drops to 0 (t=4 to t=7)

✔️ Match: Graph 3 → Third V graph

---

🔹 Graph 4 (Top-right)

Displacement vs. Time:
- From t=0 to t=4: from (0,5) to (4,7) → slope = (7 - 5)/(4 - 0) = 2/4 = +0.5 m/s
- From t=4 to t=6: from (4,7) to (6,1) → slope = (1 - 7)/(6 - 4) = (-6)/2 = -3 m/s
- From t=6 to t=7: flat → slope = 0 → 0 m/s

So:
- 0–4 s: v = +0.5 m/s
- 4–6 s: v = -3 m/s
- 6–7 s: v = 0 m/s

✔ This matches the fourth v-t graph (bottom-right):
- v = 0.5 m/s from t=0 to t=4
- Drops to -3 m/s from t=4 to t=6
- Then to 0 m/s from t=6 to t=7

✔️ Match: Graph 4 → Fourth V graph

---

✔ Final Answer Summary:

| D-T Graph | Matches V-T Graph |
|----------|-------------------|
| 1 (top-left) | Bottom-left (v = 1 m/s) |
| 2 (top-middle-left) | Bottom-middle-left (v = 3 → 0) |
| 3 (top-middle-right) | Bottom-middle-right (v = -2 → +1 → 0) |
| 4 (top-right) | Bottom-right (v = 0.5 → -3 → 0) |

All the velocity graphs are correctly matched to their corresponding displacement graphs based on slope analysis.

Let me know if you'd like help drawing them or explaining how to find slopes!