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Learn9FM0Option · Further Mechanics 1Momentum and impulseCoalescing and loss of kinetic energy
Momentum and impulse · Subtopic 3 of 4

Coalescing Particles and Lost Energy

When particles stick together they share one velocity, so conservation of momentum gives it directly. Momentum survives the crash; kinetic energy does not, and the difference has gone into heat, sound and crumpled metal.

Cambridge 9709 Paper 4Edexcel YMA01 WME02Edexcel 9FM0 OptionAQA 7367 Paper 3≈ 30 minRevision 5 · updated 2026-10-27
Section 1 of 14

Learning objectives

  • Find the common velocity when two particles coalesce.
  • Find an unknown mass or initial velocity from a coalescing collision.
  • Calculate the kinetic energy lost in a collision.
  • Understand why momentum is conserved but kinetic energy usually is not.
Pause and think

A lump of clay hits an identical lump at rest and they stick. They move off at half the original speed. Momentum is conserved. Is kinetic energy?

Section 2 of 14

Core idea

Start hereTrucks that couple

A moving railway truck crashes into a stationary one and they lock together. They roll on more slowly. Momentum survives, but some kinetic energy has gone: into heat, sound and bent metal.

The big idea

Coalescing objects move together afterwards: m1u1+m2u2=(m1+m2)vm_1u_1+m_2u_2=(m_1+m_2)v. Momentum is conserved; kinetic energy is lost.

1. Coupling

Two trucks couple together
5 m/sv = ?3000 kg2000 kgat restjoined: 5000 kg
3000(5)+0=5000v3000(5)+0=5000v, so v=3v=3 m/s.
Worked example3000 kg at 5 m/s couples to 2000 kg at rest

Find the common speed and the KE lost.

Goal
vv; ΔE\Delta E.
Strategy
Momentum, then KE before/after.
  1. 15000=5000v⇒v=315000=5000v\Rightarrow v=3
    Conservation.
  2. KE before=12(3000)(25)=37500\text{KE before}=\tfrac12(3000)(25)=37500
    Joules.
  3. KE after=12(5000)(9)=22500\text{KE after}=\tfrac12(5000)(9)=22500
    Joules.
Answer33 m/s; 15 00015\,000 J lost.
✓ CheckKE can't increase without an energy source ✓.
Kinetic energy before and after
0.511.522.53102030400kinetic energy (kJ)before 37.5after 22.515 kJ lost
Before: 12(3000)(52)=37 500\tfrac12(3000)(5^2)=37\,500 J. After: 12(5000)(32)=22 500\tfrac12(5000)(3^2)=22\,500 J. The difference, 15 00015\,000 J, becomes heat, sound and bending of the coupling.
Worked exampleHead-on and coalescing

2 kg at 5 m/s meets 3 kg at 4 m/s head-on; they stick.

Goal
Common velocity.
Strategy
Signs.
  1. 2(5)+3(−4)=−22(5)+3(-4)=-2
    Right positive.
  2. 5v=−2⇒v=−0.45v=-2\Rightarrow v=-0.4
    Together.
Answer0.40.4 m/s to the left.
✓ CheckThe heavier, faster-momentum side wins ✓.
Pause and think

Can KE increase in a collision with no energy input?

Summary

  • Common velocity after.
  • KE lost.
Key words
Coalesce
Join and move together.
HistoryHuygens and "vis viva". Christiaan Huygens noticed in the 1660s that in some collisions the total of mv2mv^2 stayed the same as well as the momentum. Leibniz later called mv2mv^2 the "living force" (vis viva), and a long argument followed about whether mvmv or mv2mv^2 was the true measure of motion. The answer: both are useful, and only one is always conserved.
Fun factWhere did the energy go? Energy is never destroyed, only changed in form. When two cars collide, the "lost" kinetic energy goes into bending metal, heating the crumple zones, the screech of the tyres and the bang you hear. A modern car's crumple zone is designed to absorb as much of it as possible, so that the passengers don't.
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