Physics I: Mechanics › Momentum › full formula sheet

m₁v1i + m₂v2i = (m₁ + m₂)vf

Say it: “in a perfectly inelastic collision the objects stick together and move as one — momentum is conserved, kinetic energy is not”

Inelastic collisions

The sticky collision: momentum conserved, kinetic energy sacrificed — where the missing energy goes, and the ballistic pendulum.

m₁, m₂ are the masses, v1i, v2i their signed initial velocities, vf the single shared final velocity. Momentum conserved; KE lost to heat/deformation.

Before this lesson: Conservation

Where it comes from

Two lumps of clay smack together and stick. A bullet buries itself in a wooden block. Cars crumple in a crash. These collisions conserve momentum — but not kinetic energy. Some motion-energy always pays for the deformation.

Before reading on: a 1500 kg car at 20 m/s rear-ends a stationary 1000 kg car and they lock together. Is the final kinetic energy equal to the initial, half of it, or something else? Estimate.

60% survives. KEi = 300,000 J; the pair moves at 12 m/s with KEf = 180,000 J. The missing 120,000 J became crumpling metal, heat, and sound — real energy, just no longer mechanical:

m₁v1i + m₂v2i = (m₁ + m₂)vfone equation (momentum), one unknown — the shared final velocitySay it: “the momenta before equal the combined mass times the final velocity”

Derivation

Conservation of momentum, with the sticking condition v1f = v2f = vf:

m₁v1i + m₂v2i
=
m₁v1f + m₂v2f
Step 1 — conserve momentum. Isolated collision: before-total equals after-total (see conservation).
=
(m₁ + m₂)vf
Step 2 — stick. Perfectly inelastic: v1f = v2f = vf. Factor it out.
vf
=
(m₁v1i + m₂v2i)/(m₁ + m₂)
Step 3 — solve. The final velocity is the mass-weighted average of the initial velocities. ∎

Maximum loss: sticking together loses the most KE of any collision outcome — the objects can't get closer to sharing one velocity than becoming one object. Partial bounces lose less.

How to use it

The procedure, every time:

  1. Confirm inelastic. “Stick,” “couple,” “embed,” “lock together” — one shared final velocity.
  2. Fix the positive direction. Signs on every initial velocity.
  3. Before-total. m₁v1i + m₂v2i (rest objects contribute 0).
  4. After = (m₁+m₂)vf. Set equal, solve for vf.
  5. Optional: the KE audit. KEi vs KEf — the difference is the energy lost to deformation/heat.
Common mistake: conserving kinetic energy too — setting ½m₁v1i² = ½(m₁+m₂)vf². Inelastic means KE is NOT conserved; only momentum is.

Worked examples

Four problems, easiest first. Momentum in, one velocity out.

Example 1 — rear-end: 1500 kg at 20 m/s sticks to stationary 1000 kg

  1. Before. 1500 × 20 = 30,000.
  2. After. 2500 × vf.
  3. Solve. vf = 12 m/s.
  4. KE audit. Before: 300,000 J. After: ½×2500×144 = 180,000 J. Lost: 120,000 J to crumpling.
Common mistake: “KE is conserved” giving vf ≈ 15.5 m/s (√(2·300000/2500)). That answer conserves energy the collision actually destroyed.
Your turn — 2000 kg at 10 m/s sticks to stationary 1000 kg. vf = ? KE lost?

Answer: vf ≈ 6.67 m/s. 20,000/3000 ≈ 6.67. KE: 100,000 → 66,667 J; lost ≈ 33,333 J.

Example 2 — ballistic pendulum: 0.02 kg bullet at 300 m/s into 2 kg block

  1. Collision (momentum). 0.02 × 300 = 2.02 × vf → vf ≈ 2.97 m/s.
  2. Swing (energy). Now the block+bullet swing up: ½(2.02)vf² = (2.02)gh → h = vf²/(2g) = 8.82/19.6 ≈ 0.45 m.
  3. Two laws, two stages. Momentum for the split-second impact (external impulse negligible); energy for the slow swing (gravity conservative).
Common mistake: using energy for the impact too — ½mv² of the bullet = (M+m)gh directly. The impact destroys KE; only momentum survives it.
Your turn — 0.01 kg at 400 m/s into 3 kg block. vf and rise h?

Answer: vf ≈ 1.33 m/s, h ≈ 0.09 m. 4/3.01 ≈ 1.33; h = 1.77/19.6 ≈ 0.09 m.

Example 3 — head-on stick: 2 kg at +6 m/s meets 3 kg at −4 m/s

  1. Before. 2×6 + 3×(−4) = 12 − 12 = 0.
  2. After. 5 × vf = 0 → vf = 0 m/s — dead stop.
  3. KE audit. Before: 36 + 24 = 60 J. After: 0. All 60 J destroyed — maximum loss, as advertised.
Common mistake: “they bounce back” — no. Perfectly inelastic means stick; the zero total momentum means the stuck lump can't move.
Your turn — 4 kg at +3 m/s meets 2 kg at −6 m/s; stick. vf = ?

Answer: 0 m/s. 12 − 12 = 0 → stuck at rest.

Example 4 — both moving: 1200 kg at 15 m/s catches 800 kg at 5 m/s, stick

  1. Before. 1200×15 + 800×5 = 18,000 + 4,000 = 22,000.
  2. After. 2000 × vf = 22,000 → vf = 11 m/s.
  3. Sanity. Between 15 and 5, closer to the heavier car's speed. ✓
Before moving on: in Example 3 all 60 J of KE vanished. Where exactly did it go — and is total energy still conserved?
Common mistake: vf = 15 m/s (“the big car wins completely”). Momentum is shared, not seized — the average, weighted by mass.
Your turn — 1000 kg at 20 m/s sticks to 1500 kg at 10 m/s (same direction). vf = ?

Answer: 14 m/s. (20,000+15,000)/2500 = 14 m/s.

Memorization tips

  • Say it aloud: “stick together: momentum conserved, kinetic energy lost.”
  • One velocity out: (m₁+m₂)vf — the combined mass is the signature. If you wrote two final velocities, it's not perfectly inelastic.
  • Mass-weighted average: vf always lands between the initial velocities, nearer the heavier mass's.
  • Never conserve KE here: the #1 error. Momentum only.
  • Ballistic pendulum = two stages: momentum for the bang, energy for the swing. Different stages, different laws.
  • Lost KE isn't lost energy: it's heat, sound, deformation. The universe's books still balance.

Final challenge

Five mixed questions — sticking, KE audits, and the two-stage pendulum. Score 5/5 and inelastic collisions are yours.

← Back to the Physics I formula sheet

How to learn a formula here

  1. Read each section in order — every section ends with a short quiz. Take it before moving on; the questions test exactly what you just read.
  2. Work the examples with the answers covered, then uncover one step at a time and compare.
  3. Finish with the final challenge — five mixed questions including the classic traps.
  4. Retake what you miss — every quiz reshuffles each attempt, and every answer explains itself.

Frequently asked questions

Where does the 'lost' kinetic energy go?

Into deforming the objects (crumpling metal, squishing clay), heat, and sound. It's still energy — just no longer organized mechanical energy. Total energy is always conserved.

What's the difference between inelastic and perfectly inelastic?

Perfectly inelastic = stick together (maximum KE loss). Inelastic (general) = bounce with some KE lost — momentum conserved, KE partially kept. Perfectly inelastic is the extreme case.

Why use momentum for the impact but energy for the swing?

The impact is violent and brief: huge internal forces, negligible external impulse → conserve momentum. The swing is slow with gravity doing work → conserve mechanical energy. Match the law to the stage.

Can all the kinetic energy vanish?

Yes — Example 3: equal and opposite momenta stick to rest, 60 J → 0. Maximum loss happens when the total momentum is zero.

Is momentum really conserved when cars crumple?

Yes — during the milliseconds of impact, external impulses (tire friction) are negligible next to the collision forces. The system is effectively isolated for the crash itself.

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