Total momentum: add the signed parts
You will be able to: Calculate a system’s total momentum using one frame and sign convention.
Can two moving carts have zero total momentum?
Two carts travel toward each other on a track. A 2 kg cart moves right at 3 m/s and a 3 kg cart moves left at 2 m/s. Both move, yet their combined momentum is zero.
A useful starting point: Momentum: mass, velocity and direction →
Words and symbols before equations
- System
- The objects we choose to analyze together.
- Total momentum P
- The vector sum of the individual momenta, in kg·m/s.
- Constituent
- One object included in the system.
- Σ (sigma)
- An instruction to add all the indicated terms.
What this picture assumes
Separate instantaneous states: cart A is 2 kg at +3 m/s, cart B is 3 kg. These bars are momenta, not spatial positions or an animation.
Connect the picture to the physics
Draw a boundary around both carts and use the floor frame for both velocities. With right positive, p_A=+6 kg·m/s and p_B=−6 kg·m/s. Add signed values: P=p_A+p_B=0.
Zero total momentum does not mean every object is at rest. Opposite vectors can cancel. The individual kinetic energies are positive and add to 15 J; energy does not cancel with direction.
The sum tells us the momentum at this moment. Whether it stays constant depends on external forces over time, which we investigate later. Always state the selected system: the first cart alone does not have zero momentum.
A worked example, step by step
A 1 kg cart travels right at 4 m/s while a 2 kg cart travels left at 1 m/s. Find the total momentum. What leftward speed of the second cart would make the total zero?
- Right positive: p_A=(1)(4)=+4 kg·m/s; p_B=(2)(−1)=−2 kg·m/s.
- P=4−2=+2 kg·m/s, pointing right.
- For zero total: 4+2v_B=0, so v_B=−2 m/s.
- Opposite momenta must have equal magnitudes; unequal masses then require unequal speed magnitudes.
Adding momentum magnitudes ignores direction and can turn a zero total into a nonzero answer.
If total momentum is zero, must total kinetic energy be zero?
Compare with an explanation
No. Two objects with equal and opposite momenta can both have positive kinetic energy.
Predict. Change one thing. Explain.
Keep the 2 kg cart at +3 m/s. Change the 3 kg cart’s velocity until total momentum is zero. Explain why the two speeds are different.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
p_A=+6; p_B=-6; P=0 kg·m/s. The two moving carts have zero total momentum.
Separate instantaneous states: cart A is 2 kg at +3 m/s, cart B is 3 kg. These bars are momenta, not spatial positions or an animation.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use a momentum or impulse relationship to justify your prediction.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionA 2 kg cart moves at +4 m/s; a 4 kg cart moves at −1 m/s. (a) Find each momentum. (b) Find P. (c) Find the second cart’s velocity needed for P=0. (d) Explain why a zero P does not establish that the objects have stopped.
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Compare with the answer and four-point rubric
- 1 point: p_A=+8 and p_B=−4 kg·m/s.
- 1 point: P=+4 kg·m/s.
- 1 point: v_B=−2 m/s.
- 1 point: Signed vectors can cancel while the objects continue moving.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1How do you add momentum in 1D?
Use signed values in one frame.
RECALL 2Can P=0 while objects move?
Yes, their momenta can cancel.
RECALL 3What must you specify before totaling?
The system, frame and positive direction.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Total momentum: add the signed parts
- P=Σp=Σmv, using the same reference frame.
- In 1D, add signed momenta. A zero sum can contain moving objects.
Remember: Adding momentum magnitudes ignores direction and can turn a zero total into a nonzero answer.
Conditions: Separate instantaneous states: cart A is 2 kg at +3 m/s, cart B is 3 kg. These bars are momenta, not spatial positions or an animation.
Refresh Kid · Unit 4 · Objectives 4.1.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 4.1, objectives 4.1.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. Fall-2026 corrections also checked. The lesson breakdown and questions are original Refresh Kid work, not official topic subdivisions.
Implementation and automated checks are separate from independent teacher review and observation of students. Both human review stages remain pending. This is a review edition, not a certified or validated assessment.
Optional further resource: College Board’s released questions and scoring guides. Papers can combine units; this link is an archive, not an assignment of every question to this lesson.
Our learn, explore, practice and recall sequence is informed by the IES learning guide. The exact Refresh Kid implementation has not been evaluated for learning effectiveness.
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