What can motion graphs and tables tell you?
You will be able to: Read graph heights and slopes and distinguish displacement from distance.
BC foundation: Unit 4 shares these contextual differentiation objectives with AB. Interpret signed rates with units, relate changing quantities before substituting an instant, and check the conditions behind approximations and limit methods. Parametric and polar motion come later.
What can motion graphs and tables tell you?
A car starts 5 meters from the origin, reaches 1 meter and returns to 5 meters. Its final coordinate equals its initial coordinate, yet it has traveled 8 meters.
A useful starting point: When is a particle speeding up, slowing down or turning? →
Words and symbols before equations
- Displacement
- Final position minus initial position.
- Distance traveled
- The total length of the path, adding each piece regardless of direction.
- Graph height
- The value of the graphed quantity.
- Secant estimate
- A finite-interval rate used to approximate a derivative.
What this picture assumes
Original mathematical model; readouts are rounded. Same exact motion rule. Known derivative v=2t−4 proves a single turn at t=2. Finite table rows alone would not prove no additional turns.
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 s: position 2 m, velocity -2 m/s, speed 2 m/s, acceleration 2 m/s². leftward; slowing down. On [0,4], displacement=0 m and distance=8 m.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
On a position graph, height is position and slope is velocity. On a velocity graph, height is velocity and slope is acceleration. Always read the axis label before interpreting a feature.
For s=t²−4t+5 on [0,4], the only turn is at t=2. The positions are s(0)=5, s(2)=1 and s(4)=5. Displacement is 0 m; distance is abs(1−5)+abs(5−1)=8 m.
This distance calculation uses a known position rule and every direction-change time. A sparse position table alone cannot rule out extra turns between samples and can underestimate travel.
If v(1.9)=−0.2 and v(2.1)=0.2 m/s, the centered acceleration estimate is (0.2−(−0.2))/0.2=2 m/s². It is exact for the stated linear v; in general finite differences are estimates.
A worked example, step by step
A known motion moves from s(0)=2 to s(1)=7 to s(3)=4 meters, with no other turns. Find displacement and distance.
- Displacement uses endpoints: 4−2=2 m.
- First leg length is abs(7−2)=5 m.
- Second leg length is abs(4−7)=3 m.
- Total distance is 8 m; the no-other-turns condition makes this sum complete.
Adding only endpoint changes misses backtracking; table rows do not prove what happened between them.
Does a horizontal velocity graph mean the car is at rest?
Compare with an explanation
Only if its height is zero. A nonzero horizontal velocity graph means constant nonzero velocity and zero acceleration.
Predict. Change one thing. Explain.
Inspect the position and velocity graphs together. Explain how the velocity zero at t=2 identifies the point where the distance calculation must split.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
t=1 s: position 2 m, velocity -2 m/s, speed 2 m/s, acceleration 2 m/s². leftward; slowing down. On [0,4], displacement=0 m and distance=8 m.
Original mathematical model; readouts are rounded. Same exact motion rule. Known derivative v=2t−4 proves a single turn at t=2. Finite table rows alone would not prove no additional turns.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the relevant rate relationship, tangent estimate, or limit argument and its conditions. 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 questionFor s=t²−4t+5 on [1,3], find displacement and distance, showing the needed split.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: v=2t−4 changes sign at t=2.
- 1 point: s(1)=2, s(2)=1 and s(3)=2 m.
- 1 point: Displacement=2−2=0 m.
- 1 point: Distance=abs(1−2)+abs(2−1)=2 m.
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 1What does velocity-graph slope represent?
Acceleration.
RECALL 2When can position differences give exact distance?
When the interval is split at every direction change.
RECALL 3What can a finite table hide?
Additional turns or changing rates between rows.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
What can motion graphs and tables tell you?
- Displacement=s(b)−s(a).
- Distance=sum of absolute position changes over intervals with no reversal.
Remember: Adding only endpoint changes misses backtracking; table rows do not prove what happened between them.
Conditions: Original mathematical model; readouts are rounded. Same exact motion rule. Known derivative v=2t−4 proves a single turn at t=2. Finite table rows alone would not prove no additional turns.
Refresh Kid · AP Calculus BC Unit 4 · Objectives CHA-3.B · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 4.2, CHA-3.B. CED effective Fall 2020, with Fall 2026 clarifications, checked September 17, 2026. Unit 4 has seven official topics. Topic 4.7 assesses 0/0 and ∞/∞ quotient forms; other indeterminate forms are excluded from the core lessons. Focused lesson titles and questions are original Refresh Kid teaching material.
Derivative units and signs are interpreted in context. Speed is the magnitude of velocity; turning requires a sign change. Related-rate equations hold at nearby times and are differentiated before snapshot values are inserted. Cone and ladder models have explicit physical domains. Tangent approximations remain estimates; error direction requires behavior on the relevant interval. L’Hôpital’s rule requires an eligible quotient form, nearby differentiability, nonzero denominator derivative and an existing finite or infinite derivative-ratio limit. A failed derivative-ratio limit is inconclusive about the original quotient.
The Organic Chemistry Tutor video creators and relevant descriptions were checked; full videos were not reviewed. Khan Academy’s unit destination was checked; its JavaScript lesson content was not fully readable by the research tool. OpenStax Sections 3.4, 4.1, 4.2 and 4.8 were consulted for conceptual cross-checking. No creator scripts, questions, diagrams or artwork were copied. Refresh Kid is not endorsed by these providers.
GitHub’s 3D website collection and its camera-control example informed optional spatial inspection. Graph geometry is generated from the stated original equations; self-hosted Three.js retains its MIT license. The optional 3D model shows the circular water surface and axial cross-section of a tip-down conical tank. The water radius and height obey the same similar-triangle ratio in 2D and 3D. Camera rotation only changes the view; signed flow and height controls describe an instantaneous state. Complete labeled 2D geometry, rates and equations remain available without WebGL.
Independent teacher review and observation of students remain pending. Checks do not certify scientific 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 to this lesson.
Learn → Explore → Practice → Review is informed by the IES learning guide; this exact implementation has not been evaluated with learners.
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