Short on time? This page is the recap. For the derivation, the energy chart and worked examples, use the full study guide.
Recap
- Compare the system just before and just after the collision.
- Momentum is conserved for an isolated system in every type of collision.
- Elastic: total kinetic energy after = total before. Individual kinetic energies can change.
- Inelastic: total kinetic energy decreases. Nonconservative contact forces transform it into internal energy, sound and permanent deformation.
- Perfectly inelastic: the objects stick and share one velocity, v_cm. This is the largest kinetic energy loss momentum allows.
Key relationships
| Relationship | When it applies |
|---|---|
| Σ mᵢuᵢ = Σ mᵢvᵢ (components) | every collision of an isolated system |
| Σ ½mᵢuᵢ² = Σ ½mᵢvᵢ² | elastic only |
| v₂ − v₁ = −(u₂ − u₁) | elastic, one dimension |
| v₁ = (m₁ − m₂)u₁/(m₁ + m₂), v₂ = 2m₁u₁/(m₁ + m₂) | elastic, one dimension, object 2 at rest |
| v = (m₁u₁ + m₂u₂)/(m₁ + m₂) | perfectly inelastic |
| K_after = P²/(2M) | perfectly inelastic (all mass moves at v_cm) |
Assumptions behind the numbers
- External impulses are negligible during the short contact.
- Masses do not change, and nothing is ejected.
- u means velocity before, v means velocity after; signs follow the stated axis.
Mistakes to avoid
- Saying momentum is lost in an inelastic collision. Only kinetic energy decreases.
- Conserving each object’s kinetic energy. Only the total, and only if elastic.
- Using energy conservation across an inelastic collision. Use momentum across it.
- Adding kinetic energies as vectors. They are scalars.
- Applying elastic formulas to an unclassified collision.
- “Sticking means everything stops.” Only if the total momentum is zero.
Quick self-check
- A 1.5 kg cart at 2.0 m/s hits and sticks to a 0.50 kg cart at rest. Find the final speed and the kinetic energy before and after. (1.5 m/s; 3.0 J before, 2.25 J after)
- Two equal masses collide elastically head-on; one was at rest and the other moved at 3.0 m/s. What happens? (They swap: 0 and 3.0 m/s)
- Total kinetic energy is 10 J before and 10 J after, but one object’s kinetic energy halved. What type of collision is it? (Elastic; energy moved to the other object)
Next: practice questions.