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Representing Motion

Representing Motion — American System Physics Revision Notes & Topical Questions

5 min read 6 parts Lesson 1 of 5 1 quiz

📚 American System Physics Hub / Topical Revision
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Lesson summary

Representing Motion: Diagrams, Position-Time Graphs and Velocity — AMERICAN SYSTEM PHYSICS notes: Draw motion diagrams, use position-time and velocity-time graphs, and distinguish distance, displacement, speed and velocity.The particle model represents an object as a single point.

  • time interval
  • speedometer
  • Instantaneous speed
  • Worked example 1
  • Worked example 2
  • Worked example 3

Part 1 of 4Motion diagrams, coordinate systems and vectors

The particle model represents an object as a single point.

Motion diagrams, coordinate systems and vectors
Motion diagram
A motion diagram is a series of images or dots showing an object's position at equal time intervals; equal spacing shows constant velocity and increasing spacing shows speeding up.
Distance is the total length of path travelled (scalar, never negative); displacement is the change in position, Δx = xf − xi (vector).
Time interval
A time interval is Δt = tf − ti.
Always include direction for vectors.
Collinear vectors in opposite directions subtract: 3 m east + 4 m west = 1 m west.
Vector subtraction A − B equals A + (−B): reverse B and add tip to tail.

Part 2 of 4Speed, velocity and position-time graphs

Speed is distance per unit time (scalar); velocity is displacement per unit time (vector).

Two cars at 25 m/s in opposite directions have the same speed but different velocities.
Average speed = total distance ÷ total time (not the average of the speeds unless equal times are spent at each); average velocity uses displacement,
  • so for a round trip it is zero even though average speed is not.
Rearrangements: d = vt and t = d/v.
Speedometer
The speedometer shows instantaneous speed, not velocity, because it gives no direction. Instantaneous speed is the magnitude of the instantaneous velocity.
For constant velocity, x = xi + vt; the slope of the position-time graph is the velocity (speed on a distance-time graph).
At rest
A horizontal line has slope 0, so the object is at rest; a negative slope means motion in the negative direction.
For a curved graph, instantaneous velocity is the slope of the tangent at that point.
On a velocity-time graph constant velocity is a horizontal line; the area between the line and the time axis is displacement (velocity × time); area below the axis counts as negative displacement.
Two objects meet when xA = xB: vAt + xA,i = vBt + xB,i.

Part 3 of 4Reading graphs step by step

To get velocity from a straight position-time graph, pick two well-separated points on the line, find Δx and Δt between them and divide: v = Δx/Δt, keeping the sign.

1
The intercept on the position axis is the starting position xi, so the line is written x = xi + vt.
2
A steeper line means a larger speed; a line sloping downward means the object moves back toward the origin.
3
To compare motion diagrams with graphs: equally spaced dots match a straight line, dots that spread out match a line that curves upward (speeding up),
  • and dots bunching up match a line flattening toward horizontal (slowing down).
4
Speed, distance and time can be linked by the triangle d = vt,
  • and the area under a velocity-time graph reproduces the same displacement,
    • (rectangle height v, width t)
  • which is a useful check on any answer.
5
When a question asks "how far" the answer is a distance or displacement with a unit in metres; "how fast" needs a speed or velocity in m/s; "how long" needs a time in seconds.
6
Writing the unit alongside every number catches many slips.

Part 4 of 4Worked examples

  1. A cyclist's position-time graph is a straight line from x = 20 m at t = 0 to x = 80 m at t = 12 s. Find the velocity and write the position equation. [3]3 marks

  2. Car A starts at x = 0 with v = 15 m/s. Car B starts at x = 90 m with v = 10 m/s in the same direction. When and where does A catch B? [3]3 marks

  3. A walker covers 1200 m in 10 min, rests for 5 min, then covers 600 m in 5 min. Find the average speed. [3]3 marks

  4. What is the difference between a speedometer reading and velocity? [2]2 marks

Key terms

TermMeaning
Particle modelRepresenting an object as a single point.
DisplacementChange in position, Δx = xf − xi; a vector.
Motion diagramSeries of dots showing position at equal time intervals; spacing shows how fast it moves.
DistanceTotal length of path travelled; scalar, never negative.
Time intervalΔt = tf − ti; scalar in seconds.
Average speedTotal distance travelled divided by total time taken.
Instantaneous speedMagnitude of the instantaneous velocity, as shown by a speedometer.
At restNot changing position with time; horizontal line on a position-time graph.

Avoid these mistakes

1
Averaging speeds instead of using total distance over total time.
2
Confusing distance with displacement.
3
Reading slope of a curved graph from two points — use the tangent for instantaneous velocity.
4
Forgetting that area below the time axis on a v-t graph is negative displacement.
5
Omitting direction when stating vectors.

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