Short on time? This page is the recap. For the derivations, graphs and worked examples, use the full study guide.
Recap
- A system is the set of objects you choose. Everything else is the surroundings.
- Total momentum P is the vector sum of the parts’ momenta. It equals the total mass times the center-of-mass velocity.
- Forces between parts of the system are internal. By Newton’s third law their impulses are equal and opposite, so they cancel in P.
- Momentum is conserved in every interaction. A chosen system keeps constant P only if the net external force on it is zero.
- In collisions and explosions, external impulses are tiny over the short contact time, so P just before = P just after.
- Quantitative work is in one or two dimensions; three-dimensional cases are qualitative only.
Key relationships
| Relationship | Units | Meaning |
|---|---|---|
| P = Σ mᵢvᵢ | kg·m/s | vector sum; use components |
| v_cm = Σ mᵢvᵢ / Σ mᵢ = P/M | m/s | whole system as one object |
| J_on A = −J_on B | N·s | third-law pair inside the system |
| dP/dt = ΣF_ext | N | slope of a P–t graph |
| ΔP = ∫ΣF_ext dt = J_ext | N·s | momentum transferred from outside |
| ΣF_ext,x = 0 ⇒ P_x before = P_x after | kg·m/s | conservation, one component |
Assumptions behind the numbers
- Masses do not change during the interaction.
- “Frictionless”, “air track” or “air table” means no horizontal external force.
- External forces are ignored only over the short interaction time, not afterwards.
Mistakes to avoid
- Adding speeds. Use signed velocities or components.
- Conserving each object’s momentum. Only the total is constant.
- Ignoring an external force. State the system and check ΣF_ext first.
- Conserving a component that has an external force (for example, the vertical component when something lands on a track).
- Thinking an explosion changes v_cm. Internal forces cannot.
- Mixing momentum and kinetic energy. Kinetic energy is often not conserved (Topic 4.4).
Quick self-check
- A 2.0 kg cart moves at +3.0 m/s and a 1.0 kg cart at −3.0 m/s. What is v_cm? (+1.0 m/s)
- Two skaters at rest push apart. The 70 kg skater moves off at +0.60 m/s. What is the 50 kg skater’s velocity? (−0.84 m/s)
- A 0.20 kg ball is dropped vertically into a 1.8 kg cart rolling at 1.0 m/s on a frictionless track. What is the cart’s new speed? (0.90 m/s; only P_x is conserved)
Next: practice questions.