How does an initial position anchor a velocity integral?
You will be able to: Recover both position coordinates from velocity and an initial position.
How does an initial position anchor a velocity integral?
A navigation log records how fast a vehicle moves east and north. To reconstruct its map location, we also need where it started.
A useful starting point: How does a vector keep two coordinate rates together? →
Words and symbols before equations
- Initial position
- The ordered pair R(a) at the reference time.
- Vector integral
- Integrate each component over the same interval.
- Displacement vector
- Net change in both coordinates.
- Constant vector
- One integration constant for each coordinate.
What this picture assumes
Original model. Coordinates have equal visual scales; angles are radians. Curves are sampled for display, while formulas determine the readouts. V=⟨2t,3⟩ m/s, R(0)=⟨1,2⟩ m. The dashed segment joins initial and current position; its coordinate changes are displacement, not necessarily distance.
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; initial (1,2) m; displacement ⟨1, 3⟩ m; final (2, 5) 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
R(t)=R(a)+∫ₐᵗ V(u)du means x(t)=x(a)+∫vₓ and y(t)=y(a)+∫vᵧ. The dummy variable u keeps the endpoint t separate.
For V=⟨2t,3⟩ and R(0)=⟨1,2⟩, integrate to get R=⟨1+t²,2+3t⟩. At t=2 the position is (5,8).
The displacement to t=2 is ⟨4,6⟩, not the final position. Check the result by differentiating and by substituting the initial time.
A worked example, step by step
V(t)=⟨2,4t⟩, R(1)=⟨3,−1⟩. Find R(3).
- Use the actual reference time 1: R(3)=R(1)+∫₁³V(t)dt.
- Horizontal change is ∫₁³2dt=4.
- Vertical change is [2t²]₁³=16.
- R(3)=⟨3+4,−1+16⟩=⟨7,15⟩.
An integral of velocity is displacement. Add the initial position in each component, and use the stated reference time.
Why are there two integration constants?
Compare with an explanation
The two initial coordinates are independent information.
Predict. Change one thing. Explain.
Move time for V=⟨2t,3⟩ and R(0)=⟨1,2⟩. Compare the initial point, final point and displacement components.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
t=1 s; initial (1,2) m; displacement ⟨1, 3⟩ m; final (2, 5) m.
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. V=⟨2t,3⟩ m/s, R(0)=⟨1,2⟩ m. The dashed segment joins initial and current position; its coordinate changes are displacement, not necessarily distance.
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 questionFor V=⟨3t²,−2⟩ and R(0)=⟨4,1⟩, find R(t), R(2), and verify the initial value.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Integrate to get R(t)=⟨4+t³,1−2t⟩.
- 1 point: R(2)=⟨12,−3⟩.
- 1 point: Differentiation gives ⟨3t²,−2⟩.
- 1 point: At t=0 the position is ⟨4,1⟩ as required.
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 anchors the integral?
The initial position vector.
RECALL 2What is a vector definite integral?
The ordered pair of component definite integrals.
RECALL 3How do you check the answer?
Differentiate and test the initial condition.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How does an initial position anchor a velocity integral?
- R(t)=R(a)+∫ₐᵗ V(u)du.
- Integrate both components.
- Verify R′=V and the initial value.
Remember: An integral of velocity is displacement. Add the initial position in each component, and use the stated reference time.
Conditions: Original model. Coordinates have equal visual scales; angles are radians. Curves are sampled for display, while formulas determine the readouts. V=⟨2t,3⟩ m/s, R(0)=⟨1,2⟩ m. The dashed segment joins initial and current position; its coordinate changes are displacement, not necessarily distance.
Refresh Kid · AP Calculus BC Unit 9 · Objectives FUN-8.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 9.5, FUN-8.A. 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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