Why is displacement a vector but distance a number?
You will be able to: Compute displacement, final position and distance with distinct integrals.
Why is displacement a vector but distance a number?
A runner goes once around a track and returns to the starting line. The finish position matches the start, but the runner has covered a positive distance.
A useful starting point: How can you tell whether planar motion is speeding up? →
Words and symbols before equations
- Displacement ΔR
- R(b)−R(a), a vector.
- Distance traveled
- Integral of speed, a nonnegative scalar.
- Magnitude of displacement
- Straight-line separation of endpoints.
- Final position
- Initial position plus displacement.
What this picture assumes
Original model. Coordinates have equal visual scales; angles are radians. Curves are sampled for display, while formulas determine the readouts. Radius-2 circle, angular rate 1 rad/s. Distance counts repeated traversals. The projection is the physical planar route; optional time height is not a third spatial coordinate.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- t=1.5708 s; point (0, 2) m; V=⟨-2, 0⟩ m/s. Speed 2 m/s; distance 3.14159 m; endpoint separation 2.82843 m. Tangent: horizontal. Increasing t is counterclockwise.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Integrating V componentwise gives displacement. Integrating √(vₓ²+vᵧ²) gives total distance; components do not cancel inside that square root.
On the radius-2 circle, one full turn has displacement ⟨0,0⟩ and distance 4π. Half a turn has displacement ⟨−4,0⟩ and distance 2π.
The magnitude of displacement is at most distance. With a table of velocity components, first compute speeds at sample times before approximating the distance integral; integrating the components instead estimates displacement.
| Feature | Displacement | Distance |
|---|---|---|
| Integral | Velocity vector | Speed |
| Type | Vector | Nonnegative scalar |
| Closed trip | Zero vector | Can be positive |
A worked example, step by step
V=⟨3,4⟩ m/s on [0,2] and R(0)=⟨1,−1⟩. Find displacement, final position and distance.
- Integrate both velocity components: ΔR=⟨6,8⟩ m.
- Add the initial position: R(2)=⟨7,7⟩ m.
- Speed=5 m/s, so distance=10 m.
- The displacement magnitude is also 10 m because the motion follows a straight line without reversal.
The magnitude of the integral of velocity generally differs from the integral of its magnitude.
Can total distance decrease as the ending time increases?
Compare with an explanation
No. It accumulates nonnegative speed.
Predict. Change one thing. Explain.
Compare half a turn and one full turn. Record distance and endpoint separation and explain which can return to zero.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
t=1.5708 s; point (0, 2) m; V=⟨-2, 0⟩ m/s. Speed 2 m/s; distance 3.14159 m; endpoint separation 2.82843 m. Tangent: horizontal. Increasing t is counterclockwise.
Read the representation: Teal shows a curve; orange marks the selected point, tangent, ray or swept region described above. Dashed segments are auxiliary comparisons. Read the model conditions and units before comparing lengths.
Original model. Coordinates have equal visual scales; angles are radians. Curves are sampled for display, while formulas determine the readouts. Radius-2 circle, angular rate 1 rad/s. Distance counts repeated traversals. The projection is the physical planar route; optional time height is not a third spatial coordinate.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the units, coordinate rates, parameter direction, tracing count or radial boundaries. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionA particle has velocity ⟨−2,0⟩ for one second then ⟨0,3⟩ for two seconds, starting at (5,1). Find displacement, final position, distance and endpoint separation.
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Compare with the answer and four-point rubric
- 1 point: Displacement is ⟨−2,6⟩.
- 1 point: Final position is (3,7).
- 1 point: Distance is 2(1)+3(2)=8.
- 1 point: Endpoint separation is √40=2√10, less than 8.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1Which integral gives a vector?
The integral of velocity.
RECALL 2Which integral gives distance?
The integral of speed.
RECALL 3Why can endpoint separation be smaller?
It measures a straight chord rather than the whole route.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Why is displacement a vector but distance a number?
- ΔR=∫V dt; R(b)=R(a)+ΔR.
- Distance=∫√(vₓ²+vᵧ²)dt.
- Magnitude of displacement≤distance.
Remember: The magnitude of the integral of velocity generally differs from the integral of its magnitude.
Conditions: Original model. Coordinates have equal visual scales; angles are radians. Curves are sampled for display, while formulas determine the readouts. Radius-2 circle, angular rate 1 rad/s. Distance counts repeated traversals. The projection is the physical planar route; optional time height is not a third spatial coordinate.
Refresh Kid · AP Calculus BC Unit 9 · Objectives FUN-8.B · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 9.6, FUN-8.B. CED effective Fall 2020 and Fall 2026 clarifications checked September 17, 2026. This unit covers BC topics 9.1–9.9. Parametric and vector motion is planar. Polar arc length, surface area and three-dimensional vector calculus are not assigned as required Unit 9 material.
Parametric slope and second derivatives retain their nonzero-denominator conditions. Speed is the magnitude of velocity; displacement and distance use different integrals. Polar coordinates allow signed radii, with distance abs(r). Polar tangents use Cartesian angular rates. Areas use squared radial boundaries with correct angular intervals, tracing counts and boundary switches; the pole is checked separately.
All focused explanations, examples, practice and models are original Refresh Kid work. OpenStax was consulted for mathematical cross-checking. Khan Academy’s destination was checked, but JavaScript lesson content was not fully readable by the research tool. Organic Chemistry Tutor video titles, creator and destinations were checked; full videos were not reviewed. No questions, diagrams or provider scripts were copied. Resources are optional; Refresh Kid is not affiliated with or endorsed by these providers.
GitHub’s 3D website collection informed optional camera and spatial inspection. The original parameter-lift model uses self-hosted Three.js with its MIT license. The teal projection is the physical circle. The vertical axis of the orange lift is time, not spatial height, and its scale is explicitly stated. No spatial arc-length calculation is made from that lift. Keyboard controls and labeled 2D alternatives remain available, without autoplay or required WebGL.
Independent teacher review and observation of students remain pending. Technical checks do not certify mathematical accuracy, accessibility or learning effectiveness. This is a review edition.
Optional official resource: Released AP Calculus BC questions and scoring guides. The archive spans multiple units; no entire exam question is assigned here.
Learn → Explore → Practice → Review is informed by the IES learning guide; this implementation has not been evaluated with learners.
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