Differentiate motion component by component
You will be able to: Differentiate a vector position function and distinguish speed change from direction change.
What if the motion bends and neither velocity component is constant?
A robot follows r(t) = (t²î + 2tĵ) m when t is inserted in seconds with the appropriate SI coefficients. Its x velocity increases while its y velocity stays constant, so the path curves.
A useful starting point: An angled launch and its apex →
Words and symbols before equations
- Vector function r(t)
- A position vector whose components depend on time.
- Componentwise derivative
- Differentiate each component while keeping the fixed unit vectors.
- Speed |v|
- Magnitude of the instantaneous velocity vector.
- Direction change
- A change in the orientation of velocity; it contributes to acceleration even at constant speed.
What this picture assumes
r(t) = kt²î + vᵧtĵ, with fixed Cartesian axes and r(0) = 0. Motion is two-dimensional. Read coordinates because screen axis scales differ.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- At 1 s: r = (1, 2) m; v = (2, 2) m/s; speed 2.828 m/s. Acceleration (2, 0) 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
Write coefficient units explicitly: r(t) = (1 m/s²)t²î + (2 m/s)tĵ. Differentiate each component: v(t) = (2 m/s²)tî + (2 m/s)ĵ. Differentiating again gives a = (2 m/s²)î. The y component of acceleration is zero even while y position changes.
At t = 1 s, velocity is (2, 2) m/s and speed is √8 m/s. At t = 2 s it is (4, 2) m/s and speed is √20 m/s. Acceleration points right; it need not point in the same direction as velocity.
Differentiate velocity components before taking their magnitude. The magnitude of acceleration is generally not the same as the time derivative of speed: one includes directional change, the other measures only how fast speed changes. The optional 3D vector view earlier explains k̂; quantitative motion work here stays within the Mechanics framework’s two-dimensional boundary.
A worked example, step by step
For r(t) = [(2 m/s)t]î + [(3 m/s²)t²]ĵ, calculate v, a and speed at t = 1 s.
- Differentiate x(t) = 2t to get vₓ = 2 m/s.
- Differentiate y(t) = 3t² to get vᵧ = (6 m/s²)t, so at 1 s v = (2, 6) m/s.
- Differentiate both velocity components: a = (0, 6) m/s².
- Speed is √(2² + 6²) = √40 ≈ 6.32 m/s, not the sum 8 m/s.
Acceleration does not always point along the path, and differentiating speed does not give the full acceleration vector.
Can an object accelerate at constant speed?
Compare with an explanation
Yes, if velocity changes direction. Constant speed is not constant velocity.
Predict. Change one thing. Explain.
Move time while keeping the path coefficients fixed. Compare the velocity components and tangent direction at each point; explain why the acceleration can stay fixed while the path bends.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
At 1 s: r = (1, 2) m; v = (2, 2) m/s; speed 2.828 m/s. Acceleration (2, 0) m/s².
r(t) = kt²î + vᵧtĵ, with fixed Cartesian axes and r(0) = 0. Motion is two-dimensional. Read coordinates because screen axis scales differ.
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 questionLet r(t) = [(3 m/s)t]î + [(1 m/s²)t²]ĵ. Derive v and a, calculate speed at 2 s, and explain whether v is parallel to a then.
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Compare with the answer and four-point rubric
- 1 point: v = (3, 2t) in m/s with t in s.
- 1 point: a = (0, 2) m/s².
- 1 point: At 2 s speed is √(3² + 4²) = 5 m/s.
- 1 point: Velocity has a nonzero x component while acceleration does not, so they are not parallel.
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 1How do you differentiate a vector in fixed Cartesian axes?
Differentiate each component separately.
RECALL 2Why can constant speed still allow acceleration?
The velocity direction can change.
RECALL 3Is quantitative 3D motion required here?
No. This Mechanics unit keeps quantitative motion in at most two dimensions; spatial vector notation is explained separately.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Differentiate motion component by component
- r = xî + yĵ; v = (dx/dt)î + (dy/dt)ĵ.
- a = (d²x/dt²)î + (d²y/dt²)ĵ.
- Speed = √(vₓ² + vᵧ²); direction changes also count as acceleration.
Remember: Acceleration does not always point along the path, and differentiating speed does not give the full acceleration vector.
Conditions: r(t) = kt²î + vᵧtĵ, with fixed Cartesian axes and r(0) = 0. Motion is two-dimensional. Read coordinates because screen axis scales differ.
Refresh Kid · AP Physics C: Mechanics Unit 1 (official Unit 1) · Objectives 1.5.A; 1.2.C · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 1.5, objectives 1.5.A; 1.2.C. 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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