---
title: "Position-Time Graph — AP Physics C Mechanics Definition"
description: "A position-time graph plots position vs. time, and its slope gives instantaneous velocity. Learn how AP Physics C tests slopes, derivatives, and concavity."
canonical: "https://fiveable.me/ap-physics-c-mechanics/key-terms/position-time-graph"
type: "key-term"
subject: "AP Physics C: Mechanics"
unit: "Unit 1"
---

# Position-Time Graph — AP Physics C Mechanics Definition

## Definition

A position-time graph plots an object's position on the vertical axis against time on the horizontal axis; in AP Physics C Mechanics, the slope of the tangent line at any point equals the instantaneous velocity, since v = dx/dt.

## What It Is

A position-time graph (often written as an x-t graph) shows where an object is at every moment. Time runs along the horizontal axis and [position](/ap-physics-c-mechanics/key-terms/position "fv-autolink") along the vertical axis. The big idea is that the graph's *shape* encodes the motion. The slope at any point is the [instantaneous velocity](/ap-physics-c-mechanics/unit-1/2-displacement-velocity-and-acceleration/study-guide/robnlCwaanT6NImP "fv-autolink"). In Physics C language, velocity is the derivative of position, so reading the slope of an x-t graph is literally taking dx/dt by eye.

That means a straight line is constant velocity, a flat horizontal line is an object at rest, and a curve means the velocity is changing, which is acceleration. Concavity tells you the sign of that acceleration. Curving upward (concave up) means acceleration is positive; curving downward means it's negative. One graph can hand you position, velocity, and the [direction](/ap-physics-c-mechanics/unit-1/1-scalars-and-vectors/study-guide/rVQeOgdT8itcgCoV "fv-autolink") of acceleration all at once, which is exactly why Topic 1.3 (Representing Motion) builds so much around it.

## Why It Matters

Position-time graphs live in Topic 1.3, Representing Motion, in [Unit 1](/ap-physics-c-mechanics/unit-1 "fv-autolink") (Kinematics). This is where [AP Physics C Mechanics](/ap-physics-c-mechanics "fv-autolink") establishes its core habit, which is treating motion quantities as derivatives and integrals of each other. The x-t graph is the most concrete version of that idea, because "slope of position equals velocity" is the first calculus relationship you use in the course. Everything that follows, from velocity-time graphs to non-constant acceleration problems, assumes you can translate between a graph, an equation x(t), and a physical description of motion. If you can sketch the velocity graph from a position graph (and vice versa), you've internalized the calculus structure the whole course runs on.

## Connections

### Velocity-time graph and area under the curve (Unit 1)

The two graphs are derivative-integral partners. The slope of the position graph gives you the [velocity](/ap-physics-c-mechanics/unit-1/4-reference-frames-and-relative-motion/study-guide/MhWvdpnoJuVbZ0WW "fv-autolink") graph, and going backward, the area under the velocity curve gives you displacement. Practice moving in both directions, because the exam tests the round trip.

### [Kinematic equations (Unit 1)](/ap-physics-c-mechanics/key-terms/kinematic-equations)

For constant acceleration, x(t) = x₀ + v₀t + ½at² is just the equation of a parabola. A curved position-time graph and the [kinematic equations](/ap-physics-c-mechanics/key-terms/kinematic-equations "fv-autolink") are the same physics in two costumes, one visual and one algebraic.

### Derivatives of position (Unit 1)

Physics C expects you to differentiate, not just spot slopes. Given x(t) as a function, take dx/dt for velocity and d²x/dt² for acceleration. The graph is the geometric picture of those derivatives, with concavity showing the sign of acceleration.

### Simple harmonic motion graphs (Unit 7)

Position-time graphs come back in [oscillations](/ap-physics-c-mechanics/unit-7 "fv-autolink"), where x(t) is a sinusoid. Reading amplitude, period, and the points of maximum speed (steepest slope) off an SHM graph is the same skill you build in Unit 1, just applied to a cosine curve.

## On the AP Exam

Multiple-choice questions hand you a position-time graph and ask you to find velocity from the slope, identify where the object is at rest, momentarily stopped, or moving in the negative direction, or pick the matching velocity-time graph. Questions like "What does the slope of a position-time graph indicate?" and "How is instantaneous velocity found on a position-time graph?" are the standard stems, and the answer always comes back to the tangent line and v = dx/dt. On free-response questions, graph translation shows up as a sketching task. You might be given x(t) as a function or a graph and asked to sketch v(t), justify the sign of acceleration from concavity, or compute an instantaneous velocity by differentiating. Watch for the classic traps. The slope gives velocity, not speed and not position, and a graph crossing the time axis means the object passed through the origin, not that it stopped.

## position-time graph vs velocity-time graph

They look similar but answer different questions. On a position-time graph, slope is velocity and the height tells you where the object is. On a velocity-time graph, slope is acceleration and the area under the curve gives displacement. The most common error is reading a position-time graph as if it were a velocity graph, like assuming a downward-sloping x-t line means slowing down when it actually means constant velocity in the negative direction.

## Key Takeaways

- The slope of a position-time graph at any point equals the instantaneous velocity, because v = dx/dt.
- A straight line on an x-t graph means constant velocity, and a horizontal line means the object is at rest.
- Curvature means acceleration is happening, and concavity gives its sign: concave up means positive acceleration, concave down means negative.
- Instantaneous velocity at a specific time is the slope of the tangent line at that point, not the slope between two distant points.
- Crossing the time axis means the object is at position zero, while a peak or valley (slope of zero) means the object is momentarily at rest.
- For constant acceleration, the position-time graph is a parabola that matches x = x₀ + v₀t + ½at².

## FAQs

### What does the slope of a position-time graph tell you?

The slope at any point equals the instantaneous velocity at that moment. In calculus terms, the slope is dx/dt. A steeper slope means a faster object, and a negative slope means motion in the negative direction.

### Does a position-time graph crossing zero mean the object stopped?

No. Crossing the time axis means the object is at position x = 0, which is just a location. The object stops only where the slope is zero, which shows up as a peak, valley, or flat section of the graph.

### How is a position-time graph different from a velocity-time graph?

On a position-time graph, slope gives velocity. On a velocity-time graph, slope gives acceleration and the area under the curve gives displacement. They're linked by calculus, since velocity is the derivative of position.

### How do you find instantaneous velocity on a position-time graph?

Draw the tangent line at the time you care about and compute its slope. If you're given x(t) as an equation instead of a picture, differentiate it and plug in the time. Both methods give v = dx/dt.

### What does a curved position-time graph mean?

A curve means the slope is changing, so the velocity is changing, which means the object is accelerating. Concave up indicates positive acceleration and concave down indicates negative acceleration. Under constant acceleration the curve is a parabola.

## Related Study Guides

- [1.3 Representing Motion](/ap-physics-c-mechanics/unit-1/3-representing-motion/study-guide/ZIECLULiWCrBlX16)

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