Push apart: recoil without creating momentum
You will be able to: Use momentum conservation for a separation and identify the energy source.
How can two objects start moving when total momentum stays zero?
Two carts sit still with a compressed spring between them. Release the spring and they move in opposite directions. The spring supplies energy; the pair’s opposing momenta still add to zero.
A useful starting point: Conservation of momentum →
Words and symbols before equations
- Recoil
- Motion opposite the momentum gained by another part of the system.
- Explosion model
- An internal interaction that pushes parts apart; it need not involve combustion.
- Stored energy
- Energy already present, such as elastic or chemical energy, that can become kinetic energy.
What this picture assumes
Both carts start at rest; A is 2 kg and leaves at −1 m/s. B’s velocity follows momentum balance. Negligible external impulse and spring mass. Different settings require different releases of stored energy.
Connect the picture to the physics
Include both carts and the negligible-mass spring. With the system initially at rest and negligible external impulse, 0=m_Av_Af+m_Bv_Bf. Rearranging gives v_Bf=−(m_A/m_B)v_Af.
Equal and opposite impulses create equal and opposite momenta, not necessarily equal speeds. If A is twice as massive as B, B moves twice as fast in the opposite direction.
Total kinetic energy can increase while total momentum stays zero. The new motion comes from stored spring energy, not from nowhere. If the whole system was moving initially, use its nonzero initial momentum; the final velocities need not point in opposite directions in that frame.
A worked example, step by step
A 2 kg cart and a 1 kg cart start at rest. After a spring release, the 2 kg cart moves at −1 m/s. Find the other velocity and total final kinetic energy.
- The selected two-cart-plus-spring system has P_i=0 and negligible external impulse.
- 0=2(−1)+1v_Bf, so v_Bf=+2 m/s.
- K_f=½(2)(1²)+½(1)(2²)=1+2=3 J.
- At least 3 J of stored energy must supply this motion; exactly 3 J is released into kinetic energy in the ideal lossless model.
Zero total momentum does not forbid an increase in kinetic energy. Identify the internal energy source.
Do unequal masses receive unequal-magnitude impulses from each other?
Compare with an explanation
No. The interaction impulses have equal magnitude and opposite direction; their velocity changes differ because their masses differ.
Predict. Change one thing. Explain.
Use a 2 kg cart A with final velocity −1 m/s. Change cart B’s mass and predict its required recoil speed. Each setting is a separate possible event; the required released energy changes with B’s mass.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
v_A=−1 m/s; v_B=2 m/s. P remains 0. Final K=3 J must come from stored energy in this initially stationary system.
Both carts start at rest; A is 2 kg and leaves at −1 m/s. B’s velocity follows momentum balance. Negligible external impulse and spring mass. Different settings require different releases of stored energy.
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 questionTwo carts initially at rest have masses 1 kg and 4 kg. The 1 kg cart leaves at +4 m/s. (a) Find its momentum. (b) Find the other cart’s velocity. (c) Find K_f. (d) Explain the energy source.
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Compare with the answer and four-point rubric
- 1 point: p_A=+4 kg·m/s.
- 1 point: 0=4+4v_B gives v_B=−1 m/s.
- 1 point: K_f=½(1)(16)+½(4)(1)=10 J.
- 1 point: Stored internal energy, such as spring energy, supplies the 10 J in a lossless model.
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 1Initially at rest, no external impulse: final P?
Zero.
RECALL 2Equal recoil momenta imply equal speeds?
Only for equal masses.
RECALL 3Does momentum conservation require constant K?
No. Stored energy can become kinetic energy.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Push apart: recoil without creating momentum
- For a pair initially at rest: m_Av_Af=−m_Bv_Bf.
- Stored energy can turn into kinetic energy while P stays constant.
Remember: Zero total momentum does not forbid an increase in kinetic energy. Identify the internal energy source.
Conditions: Both carts start at rest; A is 2 kg and leaves at −1 m/s. B’s velocity follows momentum balance. Negligible external impulse and spring mass. Different settings require different releases of stored energy.
Refresh Kid · Unit 4 · Objectives 4.3.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 4.3, objectives 4.3.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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