Worksheet for drawing distance-time graphs, featuring four problems with corresponding graph paper grids and speed values, designed for GCSE Foundation/Higher level students.

Problem: Drawing Distance-Time Graphs

The task involves drawing distance-time graphs for various scenarios. Below, I will solve each part step by step and explain the process.

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#### 1) Draw a line that represents the speeds given.

We are given three speeds:
- \( 100 \, \text{km/h} \)
- \( 2 \, \text{m/s} \)
- \( 15 \, \text{mph} \)

##### Graph 1: \( 100 \, \text{km/h} \)
- Speed: \( 100 \, \text{km/h} \)
- Units: Distance in kilometers, Time in hours.
- Slope: The slope of the line is the speed, \( 100 \, \text{km/h} \). This means for every hour, the distance increases by 100 km.
- Drawing: Start at the origin (0, 0). For every 1 unit of time (hour), move up 100 units on the distance axis. Draw a straight line with this slope.

##### Graph 2: \( 2 \, \text{m/s} \)
- Speed: \( 2 \, \text{m/s} \)
- Units: Distance in meters, Time in seconds.
- Slope: The slope is \( 2 \, \text{m/s} \). This means for every second, the distance increases by 2 meters.
- Drawing: Start at the origin (0, 0). For every 1 unit of time (second), move up 2 units on the distance axis. Draw a straight line with this slope.

##### Graph 3: \( 15 \, \text{mph} \)
- Speed: \( 15 \, \text{mph} \)
- Units: Distance in miles, Time in hours.
- Slope: The slope is \( 15 \, \text{mph} \). This means for every hour, the distance increases by 15 miles.
- Drawing: Start at the origin (0, 0). For every 1 unit of time (hour), move up 15 units on the distance axis. Draw a straight line with this slope.

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#### 2) A car travels 30 miles for 30 minutes. The car then stops at a service station for 10 minutes. The car continues for another 20 minutes at a constant speed of 45 mph. Draw the distance-time graph for the car.

##### Step-by-Step Analysis:
1. First Segment (Traveling 30 miles in 30 minutes):
- Speed: \( \frac{30 \, \text{miles}}{0.5 \, \text{hours}} = 60 \, \text{mph} \).
- Time: 0 to 30 minutes (0.5 hours).
- Distance: 0 to 30 miles.
- Graph: Draw a straight line from (0, 0) to (0.5, 30).

2. Second Segment (Stopped at Service Station for 10 minutes):
- Speed: 0 mph.
- Time: 30 to 40 minutes (0.5 to \( \frac{2}{3} \) hours).
- Distance: Remains constant at 30 miles.
- Graph: Draw a horizontal line from (0.5, 30) to \( \left( \frac{2}{3}, 30 \right) \).

3. Third Segment (Traveling for 20 minutes at 45 mph):
- Speed: 45 mph.
- Time: 40 to 60 minutes (\( \frac{2}{3} \) to 1 hour).
- Distance covered in 20 minutes: \( 45 \times \frac{20}{60} = 15 \, \text{miles} \).
- Total distance: \( 30 + 15 = 45 \, \text{miles} \).
- Graph: Draw a straight line from \( \left( \frac{2}{3}, 30 \right) \) to (1, 45).

##### Final Graph:
- Combine all three segments into a single graph.

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#### 3) A girl walks at a steady speed of 1.5 m/s for 20 seconds. She stops for 15 seconds. She then walks back to the start at 2 m/s. Draw the distance-time graph for the girl.

##### Step-by-Step Analysis:
1. First Segment (Walking at 1.5 m/s for 20 seconds):
- Speed: \( 1.5 \, \text{m/s} \).
- Time: 0 to 20 seconds.
- Distance covered: \( 1.5 \times 20 = 30 \, \text{meters} \).
- Graph: Draw a straight line from (0, 0) to (20, 30).

2. Second Segment (Stopped for 15 seconds):
- Speed: 0 m/s.
- Time: 20 to 35 seconds.
- Distance: Remains constant at 30 meters.
- Graph: Draw a horizontal line from (20, 30) to (35, 30).

3. Third Segment (Walking back at 2 m/s):
- Speed: \( 2 \, \text{m/s} \).
- Time: 35 to 55 seconds.
- Distance to cover: 30 meters (to return to the start).
- Time taken: \( \frac{30}{2} = 15 \, \text{seconds} \).
- Graph: Draw a straight line from (35, 30) to (55, 0).

##### Final Graph:
- Combine all three segments into a single graph.

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#### 4) A car drives for 1.5 hours at a steady speed of 60 mph, then stops for 30 minutes. The car then drives back to where it started at 45 mph. Draw the distance-time graph.

##### Step-by-Step Analysis:
1. First Segment (Driving at 60 mph for 1.5 hours):
- Speed: \( 60 \, \text{mph} \).
- Time: 0 to 1.5 hours.
- Distance covered: \( 60 \times 1.5 = 90 \, \text{miles} \).
- Graph: Draw a straight line from (0, 0) to (1.5, 90).

2. Second Segment (Stopped for 30 minutes):
- Speed: 0 mph.
- Time: 1.5 to 2 hours.
- Distance: Remains constant at 90 miles.
- Graph: Draw a horizontal line from (1.5, 90) to (2, 90).

3. Third Segment (Driving back at 45 mph):
- Speed: \( 45 \, \text{mph} \).
- Distance to cover: 90 miles.
- Time taken: \( \frac{90}{45} = 2 \, \text{hours} \).
- Total time: 2 to 4 hours.
- Graph: Draw a straight line from (2, 90) to (4, 0).

##### Final Graph:
- Combine all three segments into a single graph.

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Final Answers:

1. Draw lines with slopes corresponding to the given speeds.
2. Draw a graph with three segments: a line from (0, 0) to (0.5, 30), a horizontal line from (0.5, 30) to \( \left( \frac{2}{3}, 30 \right) \), and a line from \( \left( \frac{2}{3}, 30 \right) \) to (1, 45).
3. Draw a graph with three segments: a line from (0, 0) to (20, 30), a horizontal line from (20, 30) to (35, 30), and a line from (35, 30) to (55, 0).
4. Draw a graph with three segments: a line from (0, 0) to (1.5, 90), a horizontal line from (1.5, 90) to (2, 90), and a line from (2, 90) to (4, 0).

\[
\boxed{
\text{Graphs as described above.}
}
\]