doPhysics · Kinematics

Graphical Understanding of Motion

Why this chapter

Why this chapter exists

A table of numbers hides a motion; a graph shows it at a glance. From one curve you can read where a body was, how fast it moved, when it turned back and how hard it braked. Examiners know this, and many of the hardest questions on motion are questions about graphs. This chapter teaches you to read the three graphs of motion, to move from one to another, and to see through the traps they set.

You will learn

By the end of this chapter you will

Read position, velocity and acceleration from an x-t graph, including the sign of the acceleration from the bend of the curve.
Find acceleration as a slope and displacement and distance as areas on a v-t graph. Find the change in velocity as an area on an a-t graph.
Sketch the other two graphs of a motion from any one, and turn a story into a graph and a graph into a story.
Draw and explain the graphs of free fall, a bounce, a stop-and-return and a projectile, and avoid the speed-time and path traps.

Before you begin

You should know position, velocity and acceleration along a line, the three equations of motion, and free fall, from the chapter Motion in One Dimension. You should know that a derivative is a slope. The idea of an integral as an area is built here, so you need not know it already.

Already know this?

Lessons

The Position-Time Graph

Read a whole journey from one curve. Height is position, slope is velocity, the bend of the curve gives the sign of the acceleration, and the curve is never the path.

Reading the position time graphCurvature of xt graph gives sign of accelerationA graph is not a path

15 min0%

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The Velocity-Time Graph

The most useful graph of motion. Its slope is the acceleration, the area under it is the displacement, and area below the time axis counts as negative.

Slope of vt graph is accelerationArea under vt graph is displacementArea below the time axis is negative

16 min0%

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The Acceleration-Time Graph

The area under an acceleration-time graph is the change in velocity. With it the three graphs form a ladder, slope going down and area coming up.

Area under at graph is change in velocitySlope goes down area goes up

13 min0%

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One Motion, Three Graphs

The x-t, v-t and a-t graphs of one motion are three views of the same thing. Read them together, and learn to sketch the other two from any one.

One motion three graphsSketching from any one graph

15 min0%

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Graphs of Real Motions

The graphs of a ball thrown up, a bouncing ball and a car that stops and returns. Where the graphs turn, where they jump, and what each feature means.

Graphs of free fallGraphs of a bounceGraphs of a stop and return

15 min0%

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Stories, Graphs and Traps

Turn a story into a graph and a graph into a story. Then the three traps that catch most students, speed-time against velocity-time, the graphs of a projectile, and the curve taken for a path.

Sketching a graph from a storyTelling the story of a graphSpeed time is not velocity timeGraphs of a projectile

14 min0%

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Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

The Workbook

Questions to solve on paper, marked by you.

Open →

Glossary

The key words of this chapter, with the meaning this chapter gives them.

acceleration-time graph: A graph of acceleration against time, also called the a-t graph. The area under it is the change in velocity.
concave down: Bending downward like a dome, so that the slope decreases. An x-t graph that is concave down has a negative acceleration.
concave up: Bending upward like the inside of a cup, so that the slope increases. An x-t graph that is concave up has a positive acceleration.
displacement: The change in position, final position minus initial position. On a v-t graph it is the area above the time axis minus the area below it.
distance: The total length of path covered, which never decreases. On a v-t graph it is the whole area between the graph and the time axis, every part counted as positive.
free fall: Motion under gravity alone, with air resistance neglected. The acceleration is g downward at every instant.
initial velocity: The velocity at the start of the interval, written u. It is the starting value needed to climb from an a-t graph to a v-t graph.
integral: The sum of a quantity over many very small intervals, which is the area under its graph. The integral of v dt is the displacement.
jerk: The rate of change of acceleration, the slope of an a-t graph, measured in m/s³. A large jerk is felt as a jolt.
path: The line in space along which a body moves, drawn with a position coordinate, not time, on each axis. A graph against time is not a path.
position: The location of a body on a line, measured from the origin. On a position-time graph it is the height of the curve.
position-time graph: A graph of position against time, also called the x-t graph. Its height is the position and its slope is the velocity.
projectile: A body thrown into the air that then moves under gravity alone. Its horizontal velocity stays constant and its vertical motion is free fall.
sketch: A drawing of a graph that shows the correct shape of each piece and the correct values at the joins, without plotting every point.
slope: The steepness of a graph, the rise divided by the run. On a motion graph it is the rate at which the plotted quantity changes with time.
speed-time graph: A graph of speed against time. It never goes below the time axis; the area under it is the distance travelled.
tangent: The straight line that touches a curve at one point and has the same slope as the curve there. On an x-t graph its slope is the velocity at that instant.
turning point: The instant at which a body moving along a line stops and reverses. The x-t graph has a peak or a valley there and the v-t graph crosses the time axis.
velocity-time graph: A graph of velocity against time, also called the v-t graph. Its slope is the acceleration and the area under it is the displacement.