Keep momentum while accounting for collision energy change

1. Orient: Keep momentum while accounting for collision energy change

An inelastic collision reduces the system’s translational kinetic-energy subtotal while converting energy to other forms. A perfectly inelastic collision is the sticking case: both objects share one final velocity. Total momentum can still be conserved when external impulse is negligible. Total energy is not destroyed.

2. Explain and derive

For sticking, m 1 u 1+m 2 u 2=(m 1+m 2)v, so v is the initial center-of-mass velocity. Calculate this signed velocity before evaluating final kinetic energy½(m 1+m 2)v². Assuming initial and final kinetic energies are equal would impose an elastic condition that contradicts generic sticking.

The kinetic-energy decrease can be written½[m 1 m 2/(m 1+m 2)](u 1−u 2)². It depends on the relative approach speed and is nonnegative. The factor m 1 m 2/(m 1+m 2) is a useful reduced-mass shorthand, not a new physical object. For a given initial state and conserved total momentum, common final velocity removes all relative translational motion.

Not every inelastic collision sticks. Objects can separate while still converting some translational kinetic energy to internal energy. Conversely, an internal energy release can increase kinetic energy; then the source must be included in the energy ledger. The current sticking model does not determine the partition among heating, sound and deformation or infer a final temperature.

3. Work an example

With masses 2 and 3 kg approaching at+4 and−1 m/s, shared final velocity is(8−3)/5=1 m/s. Initial kinetic energy 17.5 J becomes final translational energy 2.5 J. The 15 J decrease equals½×(6/5)×5² and is accounted as other internal energy in the combined-system model.

4. Predict and investigate

Predict first: Can the stuck pair be at rest even though both incoming objects had nonzero kinetic energy?

Find mass choices giving a small or zero final velocity. Reconstruct initial and final momenta and energies separately. Compare the same masses with the elastic model; identify which conserved quantity is common to both and which distinguishes them. Retain settings and raw evidence in the lesson notebook.

5. Transfer and explain

Try a new practice case

A 2 kg object at 3 m/s sticks to a 1 kg object initially at rest. Find shared final velocity.

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

Why does observing that the objects separate after contact fail to prove the collision was elastic?

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: 4.3; 4.4. 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.