Coulomb ScatteringHyperbolic OrbitsReoriented ViewImpact ParameterRutherford Cross SectionLunar MissLunar FrameEnergy BoostCoulomb ScatteringCoulomb ScatteringHyperbolic OrbitsHyperbolic OrbitsThe trajectory from an The trajectory from an inverse square force inverse square force forms a conic section.forms a conic section.•ee < 1 ellipse < 1 ellipse•ee =1 parabola =1 parabola•ee >1 hyperbola. >1 hyperbola.The force center is at a The force center is at a focus.focus.attractive focusrcos1)1(2eearrepulsive focusa aeer1cos Reoriented ViewReoriented ViewOrient the incident axis to Orient the incident axis to horizontal. horizontal. Scattering mass forms a Scattering mass forms a hyperbolic trajectory.hyperbolic trajectory.attractive focusrrepulsive focusa aecot)2tan(2tan 111112tan22eeehmv0xb2221EJe 2222tanEJmImpact ParameterImpact ParameterThe potential is defined at The potential is defined at infinity.infinity.The impact parameter would The impact parameter would be closest approach for no be closest approach for no force.force.•Compare to angular Compare to angular momentummomentumbmvJ02021mvE hmv0xbrV2020212))((22tanbmvmvmbmv202tan2cot20mvb Rutherford Cross SectionRutherford Cross SectionScattering cross section is Scattering cross section is based on the potential.based on the potential.•Impact parameterImpact parameter•Differential impact Differential impact parameterparameterThe result is the Rutherford The result is the Rutherford scattering cross sectionscattering cross sectionbbdbd2dbd)2(sinsin22cot22020mvmvb )2(sin220dmvdbdmvd)2(sinsin224220Lunar MissLunar MissProblemProblemA spaceship of mass A spaceship of mass mm moving with velocity moving with velocity vv00 approaches the Moon.approaches the Moon.The impact parameter is The impact parameter is bb..The velocity The velocity vv00 is is perpendicular to the orbital perpendicular to the orbital velocity velocity VV of the moon. of the moon.xbmV0vfvShow that if the spaceship Show that if the spaceship passes behind the Moon it passes behind the Moon it gains kinetic energy as it gains kinetic energy as it leaves the Moon.leaves the Moon.MLunar FrameLunar FrameThe moon is much more The moon is much more massive.massive.•Lunar frame is CMLunar frame is CM•Incident and final energies Incident and final energies same in CMsame in CMGMbv2022cotcos1cos12cot2cot20mvGMmb 12sin2m1v0xbbMEnergy BoostEnergy BoostIn the Moon’s frame In the Moon’s frame the velocity the velocity components come components come from the scattering from the scattering angle.angle.In the observer’s In the observer’s frame the velocity is frame the velocity is boosted by the boosted by the moon.moon. sinˆcosˆ00vVjvivf 2200222021sinsin2cos vVvVvmKf sin2022021VvVvmKf
View Full Document
Unlocking...