SCIENCE · INVESTIGATE AND EXPLAIN

Differentiate a trajectory without losing its units

A derivative describes local change, not simply a difference between arbitrary endpoints.

Learning outcome: Compare β<0, β=0 and β>0. Retain a turning case and transform the same derivative into minute units without changing the physical motion.

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 β=.50 and t=2, x=1 m, v=4 m/s and a=6 m/s². Check each derivative directly before comparing signs.

Open the experiment

Predict

At β=.25 and t=2, which pair is velocity and acceleration?

Your prediction and reason

Experiment

Compare β<0, β=0 and β>0. Retain a turning case and transform the same derivative into minute units without changing the physical motion.

Differentiate a dimensioned trajectory

Worked teaching model · synthetic values · Supplied one-dimensional trajectory x=1−2t+βt³ in metres with t in seconds. Analytic derivatives describe this exact bounded polynomial, not inferred physical forces. Position and velocity may change sign independently.

Cubic coefficient: 0.25 m/s³ Evaluation time: 2 s

Reset keeps your prediction, captures and practice history.

Exact position and its turning behavior

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: supplied cubic (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Exact position and its turning behavior

supplied cubic

time (s)	position (m)
0	1
0.05	0.9000313
0.1	0.80025
0.15	0.7008437
0.2	0.602
0.25	0.5039063
0.3	0.40675
0.35	0.3107188
0.4	0.216
0.45	0.1227812
0.5	0.03125
0.55	-0.05840625
0.6	-0.146
0.65	-0.2313438
0.7	-0.31425
0.75	-0.3945313
0.8	-0.472
0.85	-0.5464687
0.9	-0.61775
0.95	-0.6856562
1	-0.75
1.05	-0.8105938
1.1	-0.86725
1.15	-0.9197812
1.2	-0.968
1.25	-1.011719
1.3	-1.05075
1.35	-1.084906
1.4	-1.114
1.45	-1.137844
1.5	-1.15625
1.55	-1.169031
1.6	-1.176
1.65	-1.176969
1.7	-1.17175
1.75	-1.160156
1.8	-1.142
1.85	-1.117094
1.9	-1.08525
1.95	-1.046281
2	-1
2.05	-0.9462188
2.1	-0.88475
2.15	-0.8154063
2.2	-0.738
2.25	-0.6523438
2.3	-0.55825
2.35	-0.4555312
2.4	-0.344
2.45	-0.2234687
2.5	-0.09375
2.55	0.04534375
2.6	0.194
2.65	0.3524062
2.7	0.52075
2.75	0.6992188
2.8	0.888
2.85	1.087281
2.9	1.29725
2.95	1.518094
3	1.75
3.05	1.993156
3.1	2.24775
3.15	2.513969
3.2	2.792
3.25	3.082031
3.3	3.38425
3.35	3.698844
3.4	4.026
3.45	4.365906
3.5	4.71875
3.55	5.084719
3.6	5.464
3.65	5.856781
3.7	6.26325
3.75	6.683594
3.8	7.118
3.85	7.566656
3.9	8.02975
3.95	8.507469
4	9

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.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Position	-1	m
Velocity	1	m/s
Acceleration	3	m/s²
Jerk	1.5	m/s³
Cubic coefficient when time is in minutes	54000	m/min³

Derivative coefficients carry units and change when the time coordinate is rescaled.

Differentiate each power: velocity−2+3βt² and acceleration6βt. A negative position is not negative speed, and zero velocity does not generally imply zero acceleration. With tau=t/60 in minutes the linear coefficient is−120 m/min and the cubic coefficient is216000β m/min³; convert derivative units consistently.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

Derivative coefficients carry units and change when the time coordinate is rescaled.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Differentiate a trajectory without losing its units

A derivative describes local change, not simply a difference between arbitrary endpoints.

Start with the supplied trajectory x(t)=1−2t+βt³. Position is in metres and time in seconds, so1 has units metres, the linear coefficient−2 has units m/s and β has units m/s³. A sum of unlike physical dimensions would not define a position. The derivative is the limit of a displacement-to-time ratio as the interval shrinks, provided this supplied function is smooth.

Differentiate powers: v=dx/dt=−2+3βt² and a=dv/dt=6βt. At β=.25 m/s³ and t=2 s, x=−1 m, v=1 m/s and a=3 m/s². Negative position means left of the chosen origin, not negative speed. The object can be at a negative coordinate while moving right and speeding up.

The velocity changes sign at sqrt(2/(3β)) only when β is positive. A zero velocity is a turning candidate; evaluate acceleration or neighboring velocity signs rather than assuming permanent rest. At β=0 the trajectory instead has constant negative velocity and zero acceleration. The model does not identify what force produced the prescribed path.

Change the time coordinate to tau=t/60 minutes. The same trajectory is x=1−120 tau+216000β tau³. Differentiating with respect to tau produces m/min, which must be divided by60 to compare with m/s. Coefficients cannot remain numerically unchanged when their time units change.

Changed-case check

Predict before reading the comparison. At β=.50 and t=2, x=1 m, v=4 m/s and a=6 m/s². Check each derivative directly before comparing signs.

Retained investigation

Compare β below zero, β=0 and β above zero. Retain a turning case and transform the same derivative into minute units without changing the physical motion. How does a changed coordinate origin differ from a changed time scale?

Source and evidence boundary

Conceptual reference checked30 October2026: OpenStax University Physics Volume1. All prose, examples and models are original. These are bounded supplied mathematical trajectories or synthetic observations, not observed physical experiments. No launch, impact or apparatus procedure is requested. Record assumptions, prior predictions, errors and independent checks; guided success is not independently demonstrated proficiency.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

How does a changed coordinate origin differ from a changed time scale?

Your explanation and limitations

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.

Next: Choose derivative spacing using an error budget

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.

Choose a science backup (maximum 36 MB)

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.

Export scope

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.

Choose the manifest and all parts together

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.