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Physics 1/Momentum

Momentum & collisions

Linear momentum, impulse, and conservation of momentum in collisions and recoil. Anchored to OpenStax University Physics Vol. 1, Ch. 9 (§9.1–9.4).

Before you start — give these a try

Attempting first primes your brain for the lesson — even if you miss. Nothing is graded or saved; it's just a warm-up.

A 2.0 kg2.0\ \text{kg} cart at 3.0 m/s3.0\ \text{m/s} collides with a stationary 1.0 kg1.0\ \text{kg} cart and they stick together. What is their common speed afterward (m/s)?

In a perfectly inelastic collision (objects stick together), which quantity is always conserved?

What you’ll be able to do

  • Compute momentum p=mvp = mv and impulse J=F Δt=ΔpJ = F\,\Delta t = \Delta p.
  • Apply conservation of momentum to collisions and recoil.
  • Distinguish elastic from inelastic collisions.
1

Momentum and impulse

Momentum is p=mvp = mv (a vector, kg·m/s). The impulse–momentum theorem says a force applied over a time changes momentum: J=F Δt=ΔpJ = F\,\Delta t = \Delta p. Spreading a collision over more time (crumple zones, airbags) lowers the force for the same Δp\Delta p. (OpenStax §9.1–9.2.)

2

Conservation of momentum

With no external force, the total momentum of a system is conserved: ∑pi=∑pf\sum p_i = \sum p_f. This holds in every collision and in recoil (a gun and bullet, a skater throwing a ball). (OpenStax §9.3.)

3

Elastic vs. inelastic collisions

Momentum is conserved in all collisions. Elastic collisions also conserve kinetic energy; inelastic ones don't (some KE becomes heat/sound), and in a perfectly inelastic collision the objects stick together. (OpenStax §9.4.)

Impulse–momentum theorem

J=F Δt=Δp=mvf−mvi.J = F\,\Delta t = \Delta p = m v_f - m v_i.

Intuition. A small force over a long time can change momentum as much as a big force over a short time.
Conservation of momentum

If ∑Fext=0\sum F_{\text{ext}} = 0, then ∑mivi=∑mivf\sum m_i v_i = \sum m_i v_f.

Intuition. Internal forces come in third-law pairs that cancel, so the system's total momentum can't change on its own.

Worked examples

Example 1

A 2.0 kg2.0\ \text{kg} cart at 3.0 m/s3.0\ \text{m/s} hits a stationary 1.0 kg1.0\ \text{kg} cart and they stick together. Find their common speed.

  1. 1

    Momentum before: pi=(2.0)(3.0)+(1.0)(0)=6.0 kg⋅m/sp_i = (2.0)(3.0) + (1.0)(0) = 6.0\ \text{kg·m/s}.

  2. 2

    After, the combined mass is 3.0 kg3.0\ \text{kg} moving at vv: pf=3.0 vp_f = 3.0\,v.

  3. 3

    Conserve momentum: 6.0=3.0 v⇒v=2.0 m/s6.0 = 3.0\,v \Rightarrow v = 2.0\ \text{m/s}.

Answer. v=2.0 m/sv = 2.0\ \text{m/s} (a perfectly inelastic collision).