Chapter 11: Impulse and Momentum - Summary Notes

11.1 Momentum and Impulse

  • Collision: short interaction with large impulsive force over a small time.
  • Momentum ($\vec{p}$): vector quantity, p⃗=mv⃗\vec{p} = m\vec{v}, units: kg m/s.
  • Newton's second law: F⃗=dp⃗dt\vec{F} = \frac{d\vec{p}}{dt}.
  • Kinetic energy: K=p22mK = \frac{p^2}{2m}.
  • Impulse approximation: Ignore other forces during brief collision time.
  • Impulse ($\vec{J}$): vector quantity, J⃗=∫<em>t</em>itfF⃗(t)dt\vec{J} = \int<em>{t</em>i}^{t_f} \vec{F}(t) dt, units: Ns (equivalent to kg m/s).
  • Impulse direction: same as force direction.
  • Momentum principle: Impulse causes change in momentum: Δp⃗=J⃗\Delta \vec{p} = \vec{J}.
  • One dimension: J<em>x=∫</em>t<em>it</em>fF<em>x(t)dtJ<em>x = \int</em>{t<em>i}^{t</em>f} F<em>x(t) dt = area under F</em>x(t)F</em>x(t) curve.
  • Or J<em>x=F</em>avgΔtJ<em>x = F</em>{avg} \Delta t.
  • Increasing collision time reduces average force for a given momentum change.

11.2 Conservation of Momentum

  • Law of conservation of momentum: Total momentum P⃗\vec{P} of an isolated system (F⃗net=0\vec{F}_{net} = 0) is constant (ΔP⃗=0\Delta \vec{P} = 0).
  • Interactions within the system do not change the system’s total momentum: P⃗<em>i=P⃗</em>f\vec{P}<em>i = \vec{P}</em>f.
  • Total momentum ($\vec{P}$) of N particles: P⃗=p⃗<em>1+p⃗</em>2+p⃗<em>3+⋯+p⃗</em>N=∑<em>k=1Np⃗</em>k\vec{P} = \vec{p}<em>1 + \vec{p}</em>2 + \vec{p}<em>3 + \cdots + \vec{p}</em>N = \sum<em>{k=1}^{N} \vec{p}</em>k.
  • Rate of change of total momentum: dP⃗dt=F⃗net\frac{d\vec{P}}{dt} = \vec{F}_{net}.

11.3 Collisions

  • Inelastic collision: Kinetic energy is lost, but momentum is conserved.
  • Perfectly inelastic collision: Objects stick together; move with a common final velocity. Mechanical energy is not conserved.
  • Perfectly elastic collision: Both momentum and mechanical energy are conserved.

11.4 Explosions

  • Explosion: Particles move apart after brief, intense interaction (opposite of a collision).
  • Momentum conservation applies; total momentum remains zero if the system is isolated and initially at rest.

Elastic collisions

  • Elastic collisions exhibit specific characteristics:
    • Momentum is conserved.
    • Kinetic energy is conserved.
    • Total mechanical energy is conserved.
  • The speed of approach equals the speed of separation: v<em>ii−V</em>ii=V<em>ff−v</em>ffv<em>{ii} - V</em>{ii} = V<em>{ff} - v</em>{ff}