https://file.aiquickdraw.com/mathbot/file/1725999243169byz5tn57.jpg

Let's analyze the displacement-time graph and solve the questions one by one.

### a) Work out Sarah's velocity between 10:00 and 11:00.
To find the velocity, we use the formula for velocity:

\[
\text{Velocity} = \frac{\text{Change in displacement}}{\text{Change in time}}
\]

From the graph, Sarah's displacement at 10:00 is 0 km, and at 11:00, it is 12 km. The time difference is 1 hour (from 10:00 to 11:00).

So,

\[
\text{Velocity} = \frac{12 \, \text{km}}{1 \, \text{hour}} = 12 \, \text{km/h}
\]

Sarah's velocity between 10:00 and 11:00 is **12 km/h**.

---

### b) Estimate the time that Sarah passed her home.
Sarah passes her home when the displacement is 0 km again. From the graph, after 12:00 (when her displacement is 12 km), she returns home and passes her starting point around **13:30**. This is where the displacement returns to 0.

So, the estimated time that Sarah passed her home is **13:30**.

---

### c) Work out Sarah's velocity for each of the last two stages of her journey.
The last two stages refer to:
1. The stage between 13:00 and 14:00.
2. The stage between 14:00 and 16:00.

#### Stage 1: 13:00 to 14:00
- At 13:00, Sarah's displacement is 3 km (she overshoots her home).
- At 14:00, she returns to her home (0 km displacement).
- Time taken: 1 hour.
- Change in displacement: \( 0 \, \text{km} - 3 \, \text{km} = -3 \, \text{km} \).

Velocity:

\[
\text{Velocity} = \frac{-3 \, \text{km}}{1 \, \text{hour}} = -3 \, \text{km/h}
\]

This means Sarah is traveling southward (back towards her home) at **-3 km/h**.

#### Stage 2: 14:00 to 16:00
- At 14:00, Sarah is at 0 km displacement (home).
- At 16:00, her displacement is -4 km (4 km south of her home).
- Time taken: 2 hours.
- Change in displacement: \( -4 \, \text{km} - 0 \, \text{km} = -4 \, \text{km} \).

Velocity:

\[
\text{Velocity} = \frac{-4 \, \text{km}}{2 \, \text{hours}} = -2 \, \text{km/h}
\]

This means Sarah is traveling southward at **-2 km/h**.

---

### d) Calculate Sarah’s average speed for her entire journey.
To find the average speed, we use:

\[
\text{Average speed} = \frac{\text{Total distance traveled}}{\text{Total time taken}}
\]

#### Total distance traveled:
- From 10:00 to 12:00: 12 km (northward).
- From 12:00 to 13:30: 12 km (back to home).
- From 13:00 to 14:00: 3 km (northward).
- From 14:00 to 16:00: 4 km (southward).

So, the total distance is:

\[
12 + 12 + 3 + 4 = 31 \, \text{km}
\]

#### Total time taken:
- From 10:00 to 16:00: 6 hours.

Now, we can calculate the average speed:

\[
\text{Average speed} = \frac{31 \, \text{km}}{6 \, \text{hours}} \approx 5.17 \, \text{km/h}
\]

So, Sarah’s average speed for her entire journey is approximately **5.17 km/h**.

---

Would you like further details or have any questions?

Here are some related questions to expand:
1. How would the graph look if Sarah had maintained a constant velocity during the entire journey?
2. What is the difference between speed and velocity in this context?
3. How would Sarah’s velocity change if she had returned directly to her home without overshooting it?
4. Can we determine Sarah’s acceleration from the graph? Why or why not?
5. How does the negative velocity between 13:00 and 16:00 reflect Sarah's motion?

**Tip:** Average speed considers total distance, while average velocity depends on the net displacement from the starting point.

Sarah left home at 10:00 and cycled north in a straight line. The diagram shows a displacement-time graph for her journey. a) Work out Sarah's velocity between 10:00 and 11:00. b) Estimate the time that Sarah passed her home. c) Work out Sarah's velocity for each of the last two stages of her journey. d) Calculate Sarah's average speed for her entire journey.

Learn how to calculate Sarah's velocity and average speed using a displacement-time graph. This problem involves key kinematic concepts such as velocity and average speed, making it suitable for students in Grades 7-10.

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