Momentum: mass, velocity and direction
You will be able to: Calculate signed momentum and distinguish momentum from kinetic energy.
Why do equal speeds not always mean equal momenta?
A 2 kg cart rolls right at 3 m/s. An identical cart rolls left at the same speed. They have equal kinetic energies, but their momenta point in opposite directions.
A useful starting point: Speed and velocity →
Words and symbols before equations
- Momentum p
- A vector describing mass times velocity; units kg·m/s.
- Mass m
- Measured in kilograms (kg); positive for these objects.
- Velocity v
- Speed with direction in m/s; choose right as positive.
- Vector
- A quantity with magnitude and direction. In one dimension, the sign identifies direction.
What this picture assumes
One-dimensional particle motion in the floor frame. Signed momentum bars use a common scale. Kinetic energy is reported separately in joules.
Connect the picture to the physics
Choose the frame and positive direction first. In one dimension p=mv. Multiplying a positive mass by a signed velocity preserves the velocity’s direction. A 2 kg cart at +3 m/s has p=+6 kg·m/s.
Reversing velocity reverses momentum. A value of −6 kg·m/s means 6 kg·m/s to the left, not a negative mass or a smaller magnitude than zero.
Momentum grows linearly with mass and velocity. Kinetic energy K=½mv² grows with speed squared and has no direction. An object at rest has zero momentum in that frame; another observer can measure a different velocity and momentum.
| Feature | Momentum | Kinetic energy |
|---|---|---|
| Meaning | Mass times velocity | Energy associated with motion |
| Type | Vector; signed in one dimension | Scalar; nonnegative |
| Units | kg·m/s | J |
| Double speed at fixed mass | Double magnitude | Four times the energy |
A worked example, step by step
A 3 kg cart moves left at 2 m/s. Find momentum and kinetic energy. What changes if it moves right at 4 m/s?
- Choose right positive: v=−2 m/s.
- p=3(−2)=−6 kg·m/s; K=½(3)(2²)=6 J.
- For v=+4 m/s, p=+12 kg·m/s and K=½(3)(4²)=24 J.
- The momentum reverses and doubles in magnitude; kinetic energy quadruples because speed doubled.
Use velocity, including its sign, for momentum. Do not attach a direction sign to kinetic energy.
Can an object have negative momentum and positive kinetic energy?
Compare with an explanation
Yes. A left-moving object has negative momentum when right is positive, and positive kinetic energy when its speed is nonzero.
Predict. Change one thing. Explain.
Keep mass at 2 kg and compare velocities +3 and −3 m/s. Then double mass at fixed velocity. Use the signed bars to describe what reverses and what scales.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
m=2 kg; v=3 m/s; p=6 kg·m/s; K=9 J. The comparison reverses velocity, keeping the same kinetic energy.
One-dimensional particle motion in the floor frame. Signed momentum bars use a common scale. Kinetic energy is reported separately in joules.
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 4 kg cart moves right at 2 m/s. (a) State a sign convention. (b) Find p. (c) Find p if it reverses at the same speed. (d) Explain whether its kinetic energy also reverses sign.
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Compare with the answer and four-point rubric
- 1 point: Right positive, left negative, in the floor frame.
- 1 point: p=+8 kg·m/s.
- 1 point: p=−8 kg·m/s.
- 1 point: No. K=½(4)(2²)=8 J in both cases because energy is scalar.
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 1Which way does momentum point?
In the direction of velocity.
RECALL 2Momentum unit?
kg·m/s, equivalent to N·s.
RECALL 3Double speed at fixed mass?
Momentum magnitude doubles; kinetic energy quadruples.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Momentum: mass, velocity and direction
- p=mv in one dimension; momentum unit kg·m/s.
- K=½mv² is scalar; momentum and energy are different quantities.
Remember: Use velocity, including its sign, for momentum. Do not attach a direction sign to kinetic energy.
Conditions: One-dimensional particle motion in the floor frame. Signed momentum bars use a common scale. Kinetic energy is reported separately in joules.
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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