Construct separate component ledgers for a planar encounter

1. Orient: Construct separate component ledgers for a planar encounter

Momentum is a vector, so conservation applies separately to each component in a fixed inertial frame. A northward contribution cannot be added to an eastward contribution as though both were positive numbers on one axis. Draw components, sum them by direction, and only then calculate a resultant magnitude if needed.

2. Explain and derive

The bounded example has a 2 kg object moving east at speed u and a 3 kg object moving north at speed w. They meet and stick with negligible external impulse. Eastward conservation gives 5 vx=2 u, while northward conservation gives 5 vy=3 w. Because the components are already separated, each equation has one unknown; no general coupled unknown-angle solver is required.

Final speed is√(vx²+vy²), and direction north of east follows the component ratio vy/vx with the correct quadrant. Doubling only u doubles vx but leaves vy unchanged, so both resultant speed and angle change. Doubling both incoming speeds doubles the whole final velocity vector and quadruples kinetic energy.

AP Physics 1 treats two-dimensional momentum semiquantitatively: setting up component equations and reasoning about changes are central. The present simple numeric construction supports that reasoning without demanding general simultaneous-angle solutions. The ideal sticking event does not predict collision geometry, spin or contact stresses.

3. Work an example

For u=3 m/s east and w=4 m/s north, the incoming momentum components are 6 and 12 kg m/s. Shared velocity components are 1.2 and 2.4 m/s; speed is√7.2≈2.68 m/s and direction about 63.4° north of east. Initial kinetic energy is 9+24=33 J; final translational energy is 18 J.

4. Predict and investigate

Predict first: If only the eastward input speed doubles, should the final northward component also double?

Compare inputs(3,4),(6,4) and(6,2) m/s. Sketch the component addition and predict direction changes before using output. Verify each component equation and distinguish a change in resultant direction from a change in total momentum magnitude. Retain settings and raw evidence in the lesson notebook.

5. Transfer and explain

Try a new practice case

A 2 kg eastward object at 4 m/s sticks to a 3 kg object moving north at 5 m/s. Find the shared northward velocity component.

Your numerical result (m/s) Check number

Reveal worked check (practice, not mastery)

This extra practice is not saved and does not award mastery. Record your first answer, explanation and whether you revealed help in the lesson notebook. Use the lesson’s saved question separately.

6. Reflect and retain evidence

If the objects did not stick, how would you write the two component conservation equations? Explain what additional information would be needed without trying to solve every possible final angle.

Retain the first answer, signed component sums, scalar energy totals, bounds, assumptions and one changed-case prediction. Public worked checks are practice, not independent mastery evidence. Synthetic readings are not a performed physical experiment.

Framework and source notes

Original teaching prose and numerical cases. College Board AP Physics 1 CED, Unit 4 topic mapping: 4.1; 4.3; AP1 semiquantitative boundary. Checked 30 October 2026. Two-dimensional cases use component setup and simple separate relations, not a general simultaneous-angle solver. This is a framework cross-reference, not AP authorization.