Graphing Motion

Why Graph Motion?

In Lesson 1, you learned how to describe motion using words and calculate speed with the formula s = d/t. But what if you want to see an object's entire journey at a glance: when it moved, when it stopped, when it sped up, and when it slowed down? That is where graphs come in.

A position-time graph (also called a distance-time graph) is one of the most powerful tools in physics. It plots an object's position (how far it is from a starting point) on the y-axis (vertical) against time on the x-axis (horizontal). The result is a visual story of the object's motion.

How to Read a Position-Time Graph

Every point on the line of a position-time graph tells you two things: where the object is and when it is there. But the real power of the graph comes from the shape of the line.

- A straight diagonal line going upward means the object is moving at a constant speed. It is covering equal distances in equal time intervals. The position increases steadily.
• A horizontal (flat) line means the object is at rest. Time is passing, but the position is not changing. The object has stopped.
• A steeper line means the object is moving faster. It is covering more distance in each time interval.
• A less steep (gentle) line means the object is moving slower.
• A curved line means the object's speed is changing. If the curve gets steeper over time, the object is speeding up (accelerating). If the curve flattens out, the object is slowing down (decelerating).

Walkthrough: Maria Walks to School

Let us trace a specific example. Maria leaves her house and walks to school. Here is what happens:

- Minutes 0-5: Maria walks at a steady pace. She covers 200 meters in 5 minutes (moving at 40 m/min).
• Minutes 5-7: Maria reaches a crosswalk and has to wait for the light. She stands still for 2 minutes.
• Minutes 7-10: The light changes and Maria walks faster for the last stretch. She covers 240 meters in 3 minutes (moving at 80 m/min).

On the position-time graph below, you can see all three segments clearly. The first segment is a straight line going up at a moderate slope (constant speed, 40 m/min). The second segment is a flat horizontal line (at rest, 0 m/min). The third segment is a steeper straight line going up (faster constant speed, 80 m/min).

Calculating Speed from Graphs

In Lesson 1, you learned the speed formula: s = d/t. On a position-time graph, you can calculate speed directly from the graph itself using the slope of the line.

The Slope-Speed Connection

In math, the slope of a line measures its steepness. It is calculated as "rise over run": how much the line goes up (rise) divided by how far it goes across (run). On a position-time graph:

- The rise is the change in position (distance), which we call $\Delta d$ (read as "delta d")
• The run is the change in time, which we call $\Delta t$ (read as "delta t")

So:

$$\text{slope} = \frac{\text{rise}}{\text{run}} = \frac{\Delta d}{\Delta t} = \frac{d_2 - d_1}{t_2 - t_1}$$

Notice something familiar? This is exactly the speed formula, s = d/t! The slope of a position-time graph IS the speed.

Worked Example 1: Calculating Speed from a Graph

Look at the first segment of Maria's graph (minutes 0 to 5).

Step 1: Pick two points on the line.
• Point 1: time = 0 min, position = 0 m
• Point 2: time = 5 min, position = 200 m

Step 2: Calculate slope (speed).
• speed = (200 - 0) / (5 - 0) = 200/5 = 40 m/min

Maria was walking at 40 meters per minute during the first segment.

Worked Example 2: A Steeper Line

Now look at the third segment of Maria's graph (minutes 7 to 10).

Step 1: Pick two points.
• Point 1: time = 7 min, position = 200 m
• Point 2: time = 10 min, position = 440 m

Step 2: Calculate slope (speed).
• speed = (440 - 200) / (10 - 7) = 240/3 = 80 m/min

Maria was walking at 80 m/min in the third segment, twice as fast as the first. And you can see this on the graph: the third segment is twice as steep.

Comparing Multiple Objects on One Graph

When two or more objects are plotted on the same position-time graph, you can compare their motion at a glance:

- The object with the steeper line is moving faster.
• The object with the less steep line is moving slower.
• Where the lines cross (intersect), the two objects are at the same position at the same time. In real life, this means they have met or passed each other at that moment.

The graph below shows two runners, Runner A and Runner B, starting a race from the same point.

Reading Motion Stories from Graphs

Every position-time graph tells a story. Once you know how to read the shapes, you can reconstruct exactly what happened during an object's journey, even if you were not there to see it.

Translating Graph Segments into Motion

Imagine a delivery robot leaving a warehouse. Its position-time graph has four segments:

- Segment 1 (0-3 min): A steep straight line going up. The robot is moving quickly away from the warehouse at a constant speed.
• Segment 2 (3-5 min): A horizontal flat line. The robot has stopped, perhaps making a delivery. Its position does not change.
• Segment 3 (5-8 min): A gentle straight line going up. The robot is moving again, but more slowly than before (less steep = slower speed).
• Segment 4 (8-12 min): A straight line going downward. The robot is moving back toward the warehouse. Its distance from the starting point is decreasing.

That last segment is important. A line going down on a position-time graph does not mean the object slowed down. It means the object is returning toward its starting point. The position (distance from start) is getting smaller because the object is retracing its path.

What About Curved Lines?

You may occasionally see a curved line on a position-time graph. A curve that gets steeper over time means the object is accelerating (speeding up). A curve that gets flatter over time means the object is decelerating (slowing down). You will study acceleration in detail in Lesson 3.

Real-World Graph Reading

Position-time graphs are not just a classroom exercise. They appear in many real-world applications:

- GPS and fitness apps show your running or cycling route with speed variations over time.
• Scientists track animal migration using position data from GPS collars, plotting distance traveled against days elapsed.
• Self-driving cars continuously graph their position to monitor and adjust their motion.

The skill you are building right now, translating between motion and graphs, is the same skill that engineers and scientists use every day.