SCIENCE · INVESTIGATE AND EXPLAIN

Resolve independent components and an event boundary

Use a shared event time while permitting different acceleration histories in perpendicular directions.

Learning outcome: At vertical launch velocity 2 compare horizontal jerk 0,1,3. Derive the same positive vertical root and different horizontal ranges; then change vertical velocity and explain both consequences.

Starting recap

Differentiate or integrate with stated units, initial values, event boundaries and evidence limits.

All lessons are open directly. Your saved experiments remain when you leave and return.

Try

With vy 0=2 and jx=1, landing time(2+√44)/10≈.863325 s; x≈1.83389 m.

Open the experiment

Predict

Can changing only horizontal jerk alter the ground-event time under this model?

Your prediction and reason

Experiment

At vertical launch velocity 2 compare horizontal jerk 0,1,3. Derive the same positive vertical root and different horizontal ranges; then change vertical velocity and explain both consequences.

Solve a vertical event before propagating horizontal motion

Worked teaching model · synthetic values · Original x=2t+jx t³/6,y=2+vy 0 t−5t²; ground y=0, g=10. No drag. jx=0 is a projectile; jx≠0 is prescribed nonuniform horizontal acceleration, not gravity-only projectile motion. Stop path at first positive ground event; no physical launch.

Horizontal jerk jx: 1 m/s³ Initial vertical velocity: 2 m/s

Reset keeps your prediction, captures and practice history.

Event-bounded trajectory

Axes rescale independently. Curves join sampled coordinates; a joining segment is not additional measured evidence. Read physical units rather than screen angles or apparent slopes.

Event-bounded trajectory: numerical coordinates

x (m)	y (m)
0	2
0.057559	2.05341
0.115142	2.09855
0.172772	2.1354
0.230474	2.16397
0.288271	2.18426
0.346188	2.19626
0.404247	2.19999
0.462474	2.19543
0.520891	2.1826
0.579522	2.16148
0.638392	2.13208
0.697524	2.0944
0.756941	2.04843
0.816669	1.99419
0.87673	1.93166
0.937149	1.86086
0.997949	1.78177
1.05915	1.6944
1.12079	1.59874
1.18288	1.49481
1.24544	1.3826
1.3085	1.2621
1.37209	1.13332
1.43623	0.996264
1.50094	0.850924
1.56624	0.697302
1.63217	0.535398
1.69873	0.365214
1.76597	0.186748
1.83389	0

Calculated values under the declared model
Quantity	Value	Unit
Positive ground-event time	0.863325	s
Rejected negative root	-0.463325	s
Horizontal position at event	1.83389	m
Horizontal velocity at event	2.37266	m/s
Vertical velocity at event	-6.63325	m/s
Horizontal acceleration at event	0.863325	m/s²
Vertical acceleration	-10	m/s²

Solve the vertical boundary with its own initial condition, then evaluate horizontal motion at the same event time.

Changing horizontal jerk does not change the ground-event time in this stipulated separable system. With jx=0, horizontal velocity is constant and the motion is an ideal projectile. For jx≠0 it is not gravity-only projectile motion. Negative-time intersections are outside the experiment.

Derive before checking. Retain contrasting captures and explain one assumption that could fail; calculated evidence is not actual laboratory work.

Solve the vertical boundary with its own initial condition, then evaluate horizontal motion at the same event time.

Observation to attach to a capture

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

Understand

Separate components without separating time

The supplied motion has x=2t+jx t³/6 and y=2+vy 0 t−5t². Differentiating gives vx=2+jx t²/2, ax=jx t, vy=vy 0−10t and ay=−10. The two dimensions share the same physical time while their acceleration histories differ. Horizontal acceleration is not automatically zero in every planar motion.

Identify the projectile special case

When jx=0, horizontal velocity is constant and gravity alone supplies vertical acceleration in this ideal model. When jx is nonzero, some unspecified horizontal influence produces a prescribed acceleration, so calling it an ordinary gravity-only projectile would be wrong. Component independence here is a declared mathematical boundary; drag coupling could invalidate it.

Solve the event equation

Ground contact is y=0. Solving 5t²−vy 0t−2=0 gives t=[vy 0±√(vy 0²+40)]/10. Because the square root exceeds |vy 0|, exactly one root is positive and one negative. The negative root describes an extrapolated past crossing, outside the launch-to-ground interval. Propagate x only to the positive root.

Compare the original cases

With vy 0=2, the positive time is about.863325 s regardless of jx. At jx=0 the horizontal range is 2t≈1.72665 m. At jx=1 it is 2t+t³/6≈1.83389 m. Increasing only horizontal jerk changes range and horizontal impact velocity, not the vertical event time. A curve plotted beyond ground would describe unsupported continuation through a new boundary.

Transfer and limits

For a launch 2 m high with zero vertical speed and g=8, y=2−4t² reaches zero at√.5≈.707107 s. Horizontal speed 3 then gives range about 2.12132 m, but is unnecessary for finding flight time. For different landing heights, write the new event equation rather than automatically applying a same-height range formula. Quantitative work here is two-dimensional; adding a qualitative third component does not authorize a claim of full three-dimensional dynamics.

Public numerical transfer practice

Try a new practice case

An ideal projectile begins 2 m above ground with zero vertical speed and horizontal speed 3 m/s, g=8 m/s². Find flight time.

Numerical result in the requested units

This public, unsaved check is not a secure assessment. Explain the derivative, integral or boundary in the retained course evidence workspace.

Sources and evidence boundary

Original explanations and cases, checked 30 October 2026 against the official AP Physics C Mechanics framework, OpenStax calculus kinematics and vector kinematics. All numerical records are supplied or generated, not actual laboratory measurements. No physical launch, drop or moving apparatus is requested. Public feedback is practice, not independent certification.

Apply

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

Why is a nonzero horizontal-jerk case not gravity-only projectile motion?

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.

Previous: Differentiate a moving-origin transformation Next: Design a sampling interval with competing error sources

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.