AP® Physics 1: Algebra-Based · Unit 1 · Topic 1.3
Position-Time Graphs and Their Slopes
On a position-time graph, slope equals velocity: steeper means faster, and a curving line means the velocity itself is changing, which is acceleration.
A position-time graph plots where an object is along one axis of motion against when it was there. The steepness of the line tells you how fast position is changing, which is exactly what velocity means: a lot of position change in a little time is a steep line, almost no change over a long time is a nearly flat line. A flat line isn't zero motion in general, it's zero velocity at that instant, because the object's position isn't changing.
To pull a velocity out of the graph, pick two points on the line, find the change in position (rise) and the change in time (run), and divide: v = deltax / deltat. This works perfectly when the graph is a straight line, because the slope is constant, meaning the velocity is constant. When the graph bends into a curve, the slope itself is changing from moment to moment, which means the velocity is changing, which is the definition of acceleration. A curve bending upward more steeply over time means speeding up; a curve flattening out means slowing down.
This graph is the foundation for the next two motion graphs. The slope of a position-time graph gives you the height of a velocity-time graph, and the slope of that velocity-time graph gives you acceleration. Mastering how slope and curvature translate into velocity here makes those later graphs almost automatic, since they're really the same slope idea applied one derivative further.
Key terms
- Position-time graph
- A graph showing an object's position coordinate on the vertical axis and time on the horizontal axis.
- Slope
- The rise over run of a graph, which on a position-time graph equals velocity.
- Velocity
- The rate of change of position with time, found as deltax divided by deltat.
- Acceleration
- The rate of change of velocity with time, revealed on a position-time graph by curvature rather than a constant slope.
- Curvature
- The bending of a graph's line, showing that its slope, and therefore the velocity, is changing.
Watch it explained
- How it worksWalk me through pulling a velocity out of a position-time graph using nothing but slope.
- How it worksHow does a bending position-time curve tell you the object is accelerating?
- Worked exampleFind the velocity from a straight graph running from 2 m at 0 s to 14 m at 3 s.
- Common mistakeWhy isn't a position-time graph a picture of the path the object traveled through space?
Check yourself
End of topic.