Reading and Sketching Position-Time Graphs

How would you describe exactly where something is — not "near the door," but precisely — using nothing but numbers? Legend has it that René Descartes, a 17th-century mathematician who liked to stay in bed late into the morning, was lying there one day watching a fly walk across his square-tiled ceiling, and realized he could pin down the fly's exact position with just two numbers: distance from one wall, distance from the other. Historians aren't sure the fly story is literally true — it's probably a legend attached to a real mathematician after the fact — but the invention itself is real: Descartes published this system, the coordinate plane, in 1637, and it's the reason graphs work the way they do in every math and science class since.

Imagine tracking your distance from your bedroom door over the course of a school day. Some stretches you'd be far away (in class), some you'd be right at zero (home in bed), and the line connecting those points would rise and fall as you moved around. That line is a position-time graph, and once you know how to read one, it tells you the entire story of an object's motion at a glance.

The single most useful fact about a position-time graph is this: the slope of the line equals the object's velocity. A steep slope means fast movement; a shallow slope means slow movement; a perfectly flat, horizontal line means the object isn't moving at all — it's at rest. A slope that goes downward instead of upward means the object is moving in the negative direction, back toward where it started. None of this requires numbers to see — a quick glance at how steep and which way a line tilts tells you almost everything.

Try building one from a story: someone walks away from a doorway at a steady pace, stops for a few seconds, then walks back twice as fast. On the graph, that's a straight line sloping upward, then a flat horizontal segment while they're stopped, then a steeper line sloping back down toward zero. Every feature of the motion — moving away, standing still, moving back faster — shows up as a distinct, readable feature of the graph.

There's one more thing worth noticing: what if the line isn't straight, but curves — starting shallow and getting steeper and steeper? A changing slope means the object's velocity itself is changing over time. That's the graphical fingerprint of acceleration, and it's worth being able to spot on sight even before you know how to calculate it.

Try It: Graph Runner

Play Walk the Graph to practice matching a position-time graph: set each segment's velocity so the hero follows the target. Kinematics Castle: Graph Runner

Videos

• Walking Position, Velocity and Acceleration as a Function of Time Graphs — Flipping Physics
• Motion Graphs - AP Physics 1: Unit 1 Review Supplement — Flipping Physics (YouTube)

Practice Problems

1. A skateboarder's position is 0 m at t = 0 s and 12 m at t = 4 s, moving in a straight line the whole time. Find the slope of the position-time graph (which equals the velocity).
2. Sketch a position-time graph for an object that starts at position 0 and moves at a constant velocity of 3 m/s for 5 seconds.
3. Use this position-time data table for a cart — Time (s): 0, 1, 2, 3, 4; Position (m): 0, 2, 4, 6, 8. Find the velocity (slope) between t = 0 s and t = 4 s.
4. A different cart's data table — Time (s): 0, 1, 2, 3, 4; Position (m): 0, 3, 6, 6, 6. Find the velocity during the interval from t = 0 to t = 2 s, and the velocity during the interval from t = 2 to t = 4 s. Describe in words what the object was doing during each interval.
5. Sketch a position-time graph for this story: "A person stands still for 2 seconds, then walks backward (toward negative position) at a steady pace for 3 seconds."
6. Use this position-time data table — Time (s): 0, 2, 4, 6; Position (m): 0, 10, 10, 4. a) Find the velocity for each of the three intervals (0-2 s, 2-4 s, 4-6 s). b) Which interval represents the object moving fastest, and how do you know from the numbers (not just "it looks steeper")? c) Describe, in words, what this object physically did over the full 6 seconds.
7. An object's position-time graph is a straight line with a slope of 4 m/s over a 10-second interval. If the same object had covered the exact same total displacement in only 5 seconds instead, would the slope of its new graph be steeper, less steep, or the same? Explain your reasoning first, then calculate the new slope to check.