← Motion graphs 1: position-time

Construct a position-time graph from a journey

The graph shows a museum robot's position. Which of these describes the robot's motion during segment S?

time t (s)position x (m)123456780123456S

A walker sets out from her gate along a straight sidewalk, in three stages. Stage 1: she walks forward at a steady pace, covering 8 m in 4 s. Stage 2: she stands still for 3 s. Stage 3: she walks back to the gate faster, covering the same 8 m in only 2 s.

Reading a graph turns a drawing into a journey. Constructing one runs the other way: take her journey, stage by stage, and turn it into the drawing yourself.

time t (s)position x (m)12345678902468walk outstand stillwalk back,faster
Her finished journey as a position-time graph, one stretch of time per stage.

Set up the axes first. Time runs in seconds along the horizontal axis; position runs in meters up the vertical axis. Leave room below the time axis if the journey visits negative positions. Hers never does.

Then translate each stage into a shape. A steady pace is a straight line. Standing still is a flat line. Moving in the positive direction rises; moving back falls.

Her walk out is the straight rising segment reaching 8 m at 4 s. Her stand-still is the flat segment at 8 m until 7 s. Her walk back is the falling segment returning to 0 m at 9 s.

Faster means steeper. Her return covers the same 8 m in 2 s instead of 4 s, so the falling segment is steeper than the rising one.

Join the stages end to end with no jumps: a position cannot skip, so each segment starts exactly where the last one ended.

Now you draw one. On paper, draw a position-time graph for this journey: a dog runs from its owner at a steady pace, reaching +6 m in 3 s, stands still for 2 s, then runs back to its owner in 2 s. Label both axes, then select Continue to compare your drawing with the model answer.

time t (s)position x (m)1234567802468
The model answer. Check your drawing: axes labeled time (s) and position (m); a straight rising segment reaching +6 m at t = 3 s; a flat segment at +6 m lasting 2 s, to t = 5 s; a straight falling segment returning to 0 m at t = 7 s; the falling segment steeper than the rising one, because 6 m in 2 s beats 6 m in 3 s; the three segments connected end to end with no jumps.

One more, this time starting on the negative side. On paper, draw a position-time graph for this journey: a toy train starts at −4 m, moves at a steady pace in the positive direction, reaching +4 m in 4 s, then stays put for 3 s. Label both axes, then select Continue to compare your drawing with the model answer.

time t (s)position x (m)12345678−4−20+2+4
The model answer. Check your drawing: the line starts below the time axis, at (0 s, −4 m); one single straight rising segment from −4 m to +4 m over 4 s, with no bend or pause where it crosses x = 0; a flat segment at +4 m lasting 3 s; the segments connected with no jumps, and the axes labeled time (s) and position (m).

A runner jogs in the positive direction at a steady pace, then keeps going in the positive direction twice as fast. Which graph shows the journey?

time t (s)position x (m)246804812Panel Atime t (s)position x (m)246804812Panel Btime t (s)position x (m)2468−40+4+8+12Panel Ctime t (s)position x (m)246804812Panel D

Your turn

On paper, draw a position-time graph for this journey: a remote-control car starts at +2 m, drives in the positive direction at a steady pace to +8 m in 3 s, sits still for 2 s, then drives back past its starting point to −4 m in 3 s. Label both axes, then select Continue to compare your drawing with the model answer.

time t (s)position x (m)123456789−4−20+2+4+6+8
The model answer. Check your drawing: the line starts at (0 s, +2 m), not at the origin; a straight rising segment reaching +8 m at t = 3 s; a flat segment at +8 m lasting 2 s, to t = 5 s; a straight falling segment ending at (8 s, −4 m), crossing the time axis on the way with no bend there; the falling segment steeper than the rising one, because 12 m in 3 s beats 6 m in 3 s; all segments connected with no jumps.

Starting at x = 0, a wader walks in the negative direction at a steady pace for 4 s, then stands still for 3 s. Which graph shows the journey?

time t (s)position x (m)1234567−8−40+4+8Panel Atime t (s)position x (m)1234567−8−40+4+8Panel Btime t (s)position x (m)1234567−8−40+4+8Panel Ctime t (s)position x (m)1234567−8−40+4+8Panel D

Last one, with a change of pace. On paper, draw a position-time graph for this journey: a boat leaves its dock, at position 0 m, at a steady pace in the positive direction, reaching +8 m in 4 s, then immediately returns to the dock at half that pace. Label both axes, then select Continue to compare your drawing with the model answer.

time t (s)position x (m)2468101202468
The model answer. Check your drawing: a straight rising segment reaching +8 m at t = 4 s; no flat segment at the top, because the return starts immediately; a straight falling segment returning to 0 m, shallower than the rise, because half the pace means half the steepness; the return taking 8 s, twice the outbound time, ending at t = 12 s; the segments connected with no jumps, and the axes labeled.