Differentiate the basis vectors of circular motion
Ignoring changing directions loses an acceleration term.
Learning outcome: Retain two times at the same α and two α values at one time. Derive Cartesian acceleration from the moving basis and check its magnitude.
Starting recap
Read derivatives as local rates and integrals as signed accumulation. Review power and chain rules, vector components, units and initial conditions where used.
All lessons are open directly. Your saved experiments remain when you leave and return.
Try
At α=1 and t=1, omega=1, tangential acceleration2 and inward acceleration2 m/s²; total magnitude2sqrt2.
Open the experimentPredict
The term produced by differentiating the tangent direction is:
Experiment
Retain two times at the same α and two α values at one time. Derive Cartesian acceleration from the moving basis and check its magnitude.
Differentiate moving radial and tangent directions
Worked teaching model · synthetic values · Point constrained to radius2 m with angle theta=αt²/2 from rest. The kinematic prescription gives radial and tangential acceleration; no mechanism enforcing it is supplied. Cartesian axes are inertial and spatial scales equal.
Reset keeps your prediction, captures and practice history.
Circle and evaluation radius
Spatial axes use the same pixels per physical unit, preserving geometry. 1: radius2 m path (solid).2: selected radius (dashed). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Circle and evaluation radius
radius2 m path
| east position (m) | north position (m) |
|---|---|
| 2 | 0 |
| 1.993835 | 0.1569182 |
| 1.975377 | 0.3128689 |
| 1.94474 | 0.4668907 |
| 1.902113 | 0.618034 |
| 1.847759 | 0.7653669 |
| 1.782013 | 0.907981 |
| 1.70528 | 1.044997 |
| 1.618034 | 1.175571 |
| 1.520812 | 1.298896 |
| 1.414214 | 1.414214 |
| 1.298896 | 1.520812 |
| 1.175571 | 1.618034 |
| 1.044997 | 1.70528 |
| 0.907981 | 1.782013 |
| 0.7653669 | 1.847759 |
| 0.618034 | 1.902113 |
| 0.4668907 | 1.94474 |
| 0.3128689 | 1.975377 |
| 0.1569182 | 1.993835 |
| 1.22465e-16 | 2 |
| -0.1569182 | 1.993835 |
| -0.3128689 | 1.975377 |
| -0.4668907 | 1.94474 |
| -0.618034 | 1.902113 |
| -0.7653669 | 1.847759 |
| -0.907981 | 1.782013 |
| -1.044997 | 1.70528 |
| -1.175571 | 1.618034 |
| -1.298896 | 1.520812 |
| -1.414214 | 1.414214 |
| -1.520812 | 1.298896 |
| -1.618034 | 1.175571 |
| -1.70528 | 1.044997 |
| -1.782013 | 0.907981 |
| -1.847759 | 0.7653669 |
| -1.902113 | 0.618034 |
| -1.94474 | 0.4668907 |
| -1.975377 | 0.3128689 |
| -1.993835 | 0.1569182 |
| -2 | 2.44929e-16 |
| -1.993835 | -0.1569182 |
| -1.975377 | -0.3128689 |
| -1.94474 | -0.4668907 |
| -1.902113 | -0.618034 |
| -1.847759 | -0.7653669 |
| -1.782013 | -0.907981 |
| -1.70528 | -1.044997 |
| -1.618034 | -1.175571 |
| -1.520812 | -1.298896 |
| -1.414214 | -1.414214 |
| -1.298896 | -1.520812 |
| -1.175571 | -1.618034 |
| -1.044997 | -1.70528 |
| -0.907981 | -1.782013 |
| -0.7653669 | -1.847759 |
| -0.618034 | -1.902113 |
| -0.4668907 | -1.94474 |
| -0.3128689 | -1.975377 |
| -0.1569182 | -1.993835 |
| -3.67394e-16 | -2 |
| 0.1569182 | -1.993835 |
| 0.3128689 | -1.975377 |
| 0.4668907 | -1.94474 |
| 0.618034 | -1.902113 |
| 0.7653669 | -1.847759 |
| 0.907981 | -1.782013 |
| 1.044997 | -1.70528 |
| 1.175571 | -1.618034 |
| 1.298896 | -1.520812 |
| 1.414214 | -1.414214 |
| 1.520812 | -1.298896 |
| 1.618034 | -1.175571 |
| 1.70528 | -1.044997 |
| 1.782013 | -0.907981 |
| 1.847759 | -0.7653669 |
| 1.902113 | -0.618034 |
| 1.94474 | -0.4668907 |
| 1.975377 | -0.3128689 |
| 1.993835 | -0.1569182 |
| 2 | -4.89859e-16 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
selected radius
| east position (m) | north position (m) |
|---|---|
| 0 | 0 |
| 1.080605 | 1.682942 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Angle | 1 | rad |
| Angular velocity | 1 | rad/s |
| Speed | 2 | m/s |
| Tangential acceleration | 1 | m/s² |
| Inward acceleration magnitude | 2 | m/s² |
| East acceleration | -1.922076 | m/s² |
| North acceleration | -1.14264 | m/s² |
| Total acceleration magnitude | 2.236068 | m/s² |
The derivative of the moving tangent produces inward acceleration, even when its magnitude also changes.
With er=(cosθ,sinθ) and et=(−sinθ,cosθ), der/dt=omega et and det/dt=−omega er. Therefore v=R omega et and a=R alpha et−R omega² er. The negative radial sign is not a negative acceleration magnitude. No force mechanism is inferred from this prescribed geometry.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
The derivative of the moving tangent produces inward acceleration, even when its magnitude also changes.
With er=(cosθ,sinθ) and et=(−sinθ,cosθ), der/dt=omega et and det/dt=−omega er. Therefore v=R omega et and a=R alpha et−R omega² er. The negative radial sign is not a negative acceleration magnitude. No force mechanism is inferred from this prescribed geometry.
Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.
Understand
Apply
Public practice checks a specific claim. It does not establish independent mastery.
Which step of the chain rule supplies the inward term?
Continue
Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.
Skipping retains experiments and awards no mastery.
Save a science backup or import one
Science work is stored locally in this browser, separately from mathematics and AI. It does not sync automatically. Use this science-only backup; this checkout's older all-subject export does not include science.
A legacy file is available only when the complete snapshot fits the 36 MB limit. For larger snapshots, prepare a complete file set below; never discard notebooks to make a smaller file.
Matching imported notebook drafts replace their current drafts. Omitted lessons and known first attempts remain. Export your current work first if you want to retain both drafts.
Complete file sets and course backups
A file set can hold more work than one science file. Keep its manifest and every part together. Each file is limited to 36 MB. This format needs this version of LearnAI or a compatible newer version; older versions can use only the legacy single-file backup above.
The set includes every saved science course and global continuation at this moment. Later edits need a new export. A conflicting tab exports its own visible work, which may differ from saved work in another tab.
0 file(s) selected. File order and filenames do not determine identity. Missing, changed or conflicting parts block the entire restore; no good part is applied by itself.
Science storage: loading. No account or remote computation is required.