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Understanding collisions

AshPhys · Revision cheat sheet
AS Level9702 AS Ch.6Chapter 6: MomentumLesson 6.3Sections 6.3

Goal: Tell elastic from inelastic collisions using kinetic energy or relative speeds.

velocity

01The big picture

Momentum is conserved in every collision, and so is total energy. Kinetic energy is conserved only in a perfectly elastic collision; in an inelastic one some becomes internal energy, sound or deformation.

In a perfectly elastic collision the objects separate as fast as they approached. Objects that stick together have the biggest loss of kinetic energy that conservation of momentum allows.

Perfectly elastic [paraphrase]
Total kinetic energy is the same before and after.
Inelastic [paraphrase]
Some kinetic energy is transferred to other forms.
Fig. 1 · Equal masses, perfectly elastic
Equal masses, perfectly elasticBefore: ball A moves right at 4.0 m/s towards an identical ball B at rest. After: A is at rest and B moves right at 4.0 m/s.beforeafterAB4.0 m s⁻¹at restAB4.0 m s⁻¹at rest
A stops and B moves off at A’s speed: approach speed = separation speed = 4.0 m s⁻¹.

02Key equations

Kinetic energy
Ek=12mv2E_\mathrm{k} = \tfrac{1}{2}mv^{2}
Perfectly elastic: approach = separation
u1−u2=v2−v1u_1 - u_2 = v_2 - v_1
Symbols and units
  • EkE_\mathrm{k}kinetic energyJ\mathrm{J}
  • uuvelocity beforem s−1\mathrm{m\,s^{-1}}
  • vvvelocity afterm s−1\mathrm{m\,s^{-1}}

03Sample problems with solutions

Sample 1 · Is it elastic?
A 2.0 kg trolley at 3.0 m s⁻¹ hits a stationary 1.0 kg trolley. They stick together and move at 2.0 m s⁻¹. Show that the collision is inelastic and find the kinetic energy lost.

  1. Kinetic energy before and after:

    Ek=12(2.0)(3.0)2=9.0 J,Ek′=12(3.0)(2.0)2=6.0 JE_\mathrm{k} = \tfrac{1}{2}(2.0)(3.0)^{2} = 9.0\ \mathrm{J}, \quad E_\mathrm{k}' = \tfrac{1}{2}(3.0)(2.0)^{2} = 6.0\ \mathrm{J}

  2. Kinetic energy fell, so the collision is inelastic:

    ΔEk=9.0−6.0=3.0 J\Delta E_\mathrm{k} = 9.0 - 6.0 = 3.0\ \mathrm{J}
Answer: Inelastic; 3.0 J of kinetic energy lost
Unit check: kg×(m s−1)2=J ✓\mathrm{kg} \times (\mathrm{m\,s^{-1}})^{2} = \mathrm{J}\ \checkmark

04Practice problems

  1. P1
    Which quantities are conserved in every collision?
    Final answerMomentum and total energy
    Answer: Momentum and total energy
  2. P2
    A 0.10 kg ball hits a wall at 8.0 m s⁻¹ and rebounds at 6.0 m s⁻¹. Find the kinetic energy lost.
    Final answer1.4 J
    Answer: 1.4 J
  3. P3
    Two balls approach at a relative speed of 5.0 m s⁻¹ and separate at 3.0 m s⁻¹. Is the collision perfectly elastic?
    Final answerNo: inelastic
    Answer: No: inelastic
  4. P4
    A 1.0 kg ball at 6.0 m s⁻¹ hits a stationary 2.0 kg ball elastically. Find both velocities after.
    Final answer1.0 kg: 2.0 m s⁻¹ back; 2.0 kg: 4.0 m s⁻¹ forwards
    Answer: 1.0 kg: 2.0 m s⁻¹ back; 2.0 kg: 4.0 m s⁻¹ forwards

05Other useful information

Common mistakes

  • ✗ Wrong: Momentum is lost in an inelastic collision.

    ✓ Right: Momentum is always conserved; only kinetic energy is lost.

Exam tips

  • Elastic problems: use momentum conservation plus approach speed = separation speed.

What the syllabus asks

  • 9702 topic 3: elastic and inelastic collisions; relative speeds. Coursebook 6.3.

Explore: Practice questions for this lesson