Estimate motion from measured data
You will be able to: Estimate a local slope, linearize a constant-acceleration model and discuss measurement limitations.
How can a video reveal velocity without a speed sensor?
Video frames show a cart at 0, 1, 4 and 9 m at 0, 1, 2 and 3 s. The widening gaps suggest increasing speed. These illustrative numbers follow x = t² in SI units.
A useful starting point: When acceleration changes with time →
Words and symbols before equations
- Sampling interval
- Time between successive recorded measurements.
- Central difference
- Estimate a derivative at t with [x(t+h) − x(t−h)]/(2h).
- Linearization
- Transform variables so a predicted relationship becomes a straight line.
- Residual
- Measured value minus the model prediction at that time.
What this picture assumes
Synthetic exact samples at 0, 1, 2 and 3 s, released from rest. The offset is common to every position, not random noise. Real experiments need uncertainty analysis.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- Positions at 0, 1, 2, 3 s: 0, 1, 4, 9 m. Central slope at 2 s = 4 m/s. Linearized slope = 1 m/s², so a = 2 m/s².
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
Calibrate a visible length scale and use frame timestamps. Track the same point on the object. Camera perspective, uncertain endpoints and timing errors can affect estimates, so repeated trials and a fixed perpendicular camera help.
At 2 s, the central slope estimate from 1 s and 3 s is (9 − 1)/2 = 4 m/s. This is exactly the derivative for an ideal quadratic, but only an approximation for general motion and noisy data. Shorter intervals can improve time resolution while making position noise more influential after division by a smaller time.
For release from rest at x₀ = 0 under constant acceleration, x = ½at². Plot x vertically versus t² horizontally: the slope is a/2, in m/s². If initial velocity is nonzero, the extra v₀t term invalidates that simple straight-line model. Inspect residuals and uncertainty; agreement at a few points does not prove the law universally.
A worked example, step by step
A release-from-rest experiment gives a best-fit slope of 1.5 m/s² on a graph of x against t². Find acceleration and predicted position at 2 s.
- The model is x = ½at² with x₀ = 0 and v₀ = 0.
- The graph’s slope is ½a, not a.
- Therefore a = 2(1.5) = 3.0 m/s².
- At 2 s, x = 1.5(4) = 6 m. Check residuals and the release conditions before trusting the model.
A smooth best-fit line is not evidence that measurement uncertainty vanished.
Why avoid estimating slope from two almost identical noisy positions?
Compare with an explanation
Their difference may be dominated by measurement error, which is magnified when divided by a very small time interval.
Predict. Change one thing. Explain.
Change the common position offset. Predict whether the estimated velocity or the slope of x against t² changes. Distinguish a calibration offset from random measurement noise.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Positions at 0, 1, 2, 3 s: 0, 1, 4, 9 m. Central slope at 2 s = 4 m/s. Linearized slope = 1 m/s², so a = 2 m/s².
Synthetic exact samples at 0, 1, 2 and 3 s, released from rest. The offset is common to every position, not random noise. Real experiments need uncertainty analysis.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant position, velocity, acceleration or integral relationship to justify your prediction.
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 questionDesign a video test of constant acceleration from rest. State the measured quantities, graph choice, extraction of acceleration and one limitation with a practical improvement.
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Compare with the answer and four-point rubric
- 1 point: Measure positions using a length calibration and times from frames; release from rest.
- 1 point: Plot displacement from the initial position versus elapsed time squared.
- 1 point: Use a = 2 × fitted slope, with units m/s².
- 1 point: For example: perspective distorts positions; place a fixed camera perpendicular to the motion and calibrate in the motion plane. Repeat trials and inspect residuals.
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 is a central difference?
A slope across equal time intervals on either side of the target instant.
RECALL 2What must hold for Δx vs t² slope to equal a/2?
Constant acceleration and zero initial velocity.
RECALL 3Does a constant position offset change velocity estimates?
No. Position differences remove that offset.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Estimate motion from measured data
- v(t) ≈ [x(t+h) − x(t−h)]/(2h).
- From rest at x₀ = 0: x vs t² slope = a/2.
- Report axes, units, assumptions and uncertainty sources.
Remember: A smooth best-fit line is not evidence that measurement uncertainty vanished.
Conditions: Synthetic exact samples at 0, 1, 2 and 3 s, released from rest. The offset is common to every position, not random noise. Real experiments need uncertainty analysis.
Refresh Kid · AP Physics C: Mechanics Unit 1 (official Unit 1) · Objectives 1.3.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 1.3, objectives 1.3.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. This is Mechanics Unit 1: Kinematics. Quantitative motion problems stay in one or two dimensions; the optional spatial view clarifies vector notation. Students should study calculus alongside this course. The lesson breakdown and questions are original Refresh Kid work, not official topic subdivisions.
Implementation and automated checks are separate from independent teacher review and observation of students. Both human review stages remain pending. This is a review edition, not a certified or validated assessment.
Optional further resource: College Board’s released questions and scoring guides. Papers can combine units; this link is an archive, not an assignment of every question to this lesson.
Our learn, explore, practice and recall sequence is informed by the IES learning guide. The exact Refresh Kid implementation has not been evaluated for learning effectiveness.
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