Add momenta and locate the system’s motion

1. Orient: Add momenta and locate the system’s motion

Linear momentum is the vector p=mv. In one dimension its sign follows the chosen positive axis, unlike kinetic energy, which depends on speed squared. Momentum units are kg m/s, equivalent to N s. A larger mass can have the same momentum as a smaller mass moving faster, but their kinetic energies need not match.

2. Explain and derive

Total system momentum is the vector sum of the members’ momenta. Add signed components in a common frame. Two moving objects can have zero total momentum when their contributions cancel. This does not imply zero total kinetic energy: each moving object contributes a positive kinetic-energy amount.

For fixed total mass M, the center-of-mass velocity is ptotal/M. This connects the weighted-position definition to motion. If there is no net external force, ptotal and the center-of-mass velocity remain constant even while internal interactions change individual velocities. The center can remain at rest while members approach or separate.

Changing to an observer moving at velocity u changes each momentum by−mᵢu, so total momentum changes by−Mu. In the center-of-mass frame the total is zero. Momentum conservation remains valid for an isolated system in any inertial frame, provided every before-and-after term uses that same frame.

3. Work an example

A 2 kg object at+3 m/s has momentum+6 kg m/s; a 3 kg object at−2 m/s has momentum−6 kg m/s. Total momentum and center-of-mass velocity are zero, but total kinetic energy is 9+6=15 J. Setting both velocities to+2 m/s instead gives total momentum 10 kg m/s and center velocity 2 m/s.

4. Predict and investigate

Predict first: Can total momentum vanish while every object is moving? Can total kinetic energy vanish in that same case?

Find two control pairs giving zero total momentum and compare their kinetic energies. Then choose equal velocities and verify the center has that same velocity. Explain why adding speeds instead of signed momenta would lose the system description. Retain settings and raw evidence in the lesson notebook.

5. Transfer and explain

Try a new practice case

A 2 kg object moves at+4 m/s and a 3 kg object at−1 m/s. Find center-of-mass 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

For equal momentum magnitudes, compare the kinetic energies of masses m and 2 m. Derive K=p²/(2 m) before deciding which is larger.

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. 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.