physics 321 hour 37 collisions in three dimensions
TRANSCRIPT
Physics 321
Hour 37Collisions 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.
Θ
Θ
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Collision in the CM Frame• Assume we know any energy loss as well.
[ 00𝑃0]+[ 00−𝑃0
]=[ 𝑃 s∈Θ0𝑃 cos Θ ]+[ −𝑃 s∈Θ0−𝑃 cos Θ ]
• We can find the kinetic energies is we know the momenta and the masses.
𝑇 01+𝑇02=𝑇1+𝑇 2+𝐸𝑒𝑥
Collision in the CM Frame• First, we find P.
• Then we find T1 and T2.• Now we know everything!
Elastic Scattering• If Eex=0,
• 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
𝑚1[ 00𝑉 10]+𝑚2[ 0
0−𝑉 20
]=𝑚1[ 𝑉 1 s∈Θ0
𝑉 1 co sΘ]+𝑚2[ −𝑉 2 s∈Θ
0−𝑉 2 cos Θ
][ 00𝑃0]+[ 00−𝑃0
]=[ 𝑃 s∈Θ0𝑃 cos Θ ]+[ −𝑃 s∈Θ0−𝑃 cos Θ ]
CM to Lab• So we can find the lab quantities
𝑚1[ 00
𝑉 10+𝑉 20]+𝑚2[ 000]=𝑚1[ 𝑉 1 s∈Θ
0𝑉 1co sΘ+𝑉 20
]+𝑚2[ −𝑉 2 s∈Θ0
−𝑉 2 co sΘ+𝑉 20]
[ 00
𝑃0+𝑚1
𝑚2
𝑃0]+[ 000]=[ 𝑃 s∈Θ
0
𝑃 co sΘ+𝑚1
𝑚2
𝑃0]+[ − 𝑃 s∈Θ
0−𝑃 co sΘ+𝑃0
][ 00𝑝0]+[000 ]=[ 𝑝1 s∈𝜃10
𝑝1 cos𝜃1]+[𝑝2sin 𝜃20𝑝2 cos𝜃2]
CM to Lab
[ 00
𝑃0+𝑚1
𝑚2
𝑃0]+[ 000]=[ 𝑃 s∈Θ
0
𝑃 co sΘ+𝑚1
𝑚2
𝑃0]+[ − 𝑃 s∈Θ
0−𝑃 co sΘ+𝑃0
][ 00𝑝0]+[000 ]=[ 𝑝1 s∈𝜃10
𝑝1 cos𝜃1]+[𝑝2sin 𝜃20𝑝2 cos𝜃2]
CM to LabBut usually we don’t know Θ a priori.
This is a quadratic equation we can solve for p1. Then we can find Θ and everything else.
ExampleCollisionKinematics.nb