Physics 321 Hour 37 Collisions in Three Dimensions
advertisement
Physics 321 Hour 37 Collisions in Three Dimensions Collision in the CM Frame • Assume we know the masses, the initial momentum of each body (they’re the same), and the scattering angle. π Θ π0 π0 Θ π Collision in the CM Frame • Assume we know any energy loss as well. 0 0 π sin Θ −π sin Θ 0 + 0 = + 0 0 π0 −π0 π cos Θ −π cos Θ • We can find the kinetic energies is we know the momenta and the masses. π01 + π02 = π1 + π2 + πΈππ₯ Collision in the CM Frame • First, we find P. π1 + π2 = π01 + π02 − πΈππ₯ π2 π2 π2 + = = π01 + π02 − πΈππ₯ 2π1 2π2 2π π= 2π π01 + π02 − πΈππ₯ • Then we find T1 and T2. • Now we know everything! Elastic Scattering • If Eex=0, π= 2π π01 + π02 = π0 π1 = π01 π2 = π02 • All the scattering can do is change the directions of the particles. The CM Frame • The cm frame is really easy… • Theorists (almost) always work in the cm frame. • Experimentalists usually convert cross sections to the cm as well. • But, experiments are not usually conducted in the cm frame! CM to Lab • Now let’s assume we have the CM solution and we want to get back to the lab frame. • First, we write everything in terms of velocity π1 0 Θ−π sin Θ−π2 sin Θ 1 sin 0 000 0 π sinπΘ 0 π+π 00 = 0+ + 0π2= 0 + 2 0 0=+ππ10 π 1 1 2+ π2 0 = π0 π −π −π20 Θ−π cos Θ−π2 cos Θ π cosπ−π 0 1Θcos π20+ π20 10 +π10 200+ π20 π1 sin Θ −π2 sin Θ 0 0 π1 + π2 π1 cos Θ + π20 −π2 cos Θ + π20 0 0 π10 CM to Lab • So we can find the lab quantities π1 π1 sin Θ −π2 sin Θ 0 0 0 0 0 + π2 0 = π1 + π2 π10 + π20 π1 cos Θ + π20 −π2 cos Θ + π20 0 0 π sin Θ −π sin Θ 0 0 0 0 + = + 0 π1 π1 π0 + π0 π cos Θ + π0 −π cos Θ + π0 0 π2 π2 π1 sin π1 π2 sin π2 0 0 0 + 0 = 0 0 + π0 π1 cos π1 π2 cos π2 0 CM to Lab 0 π sin Θ −π sin Θ 0 0 0 0 + 0 = + π1 π1 π0 + π0 π cos Θ + π0 −π cos Θ + π0 0 π2 π2 π1 sin π1 π2 sin π2 0 0 0 + 0 = 0 0 + π0 π1 cos π1 π2 cos π2 0 π1 π0 = π0 π π sin Θ = π1 sin π1 π1 π cos Θ + π0 = π1 cos π1 π2 π sin Θ tan π1 = π1 π cos Θ + π π2 0 π1 2 = π sin Θ 2 + 2 π1 π cos Θ + π0 π2 CM to Lab But usually we don’t know Θ a priori. π= π π0 = π0 π1 2π π01 + π02 − πΈππ₯ π sin Θ = π1 sin π1 π1 π cos Θ = π1 cos π1 − π0 π2 2 π1 π1 = π sin Θ + π cos Θ + π0 π2 2 2π π π 1 0 1 2 2 =π + π cos Θ + π π2 π2 2 0 2 2π π π π 1 0 1 1 2 2 =π + π1 cos π1 − π0 + π 0 2 π2 π2 π2 This is a quadratic equation we can solve for p1. Then we can find Θ and everything else. 2 2 Example CollisionKinematics.nb
