Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Homework Help
Advanced Physics Homework Help
Proving momentum equation for neutrino/nucleon scattering
Reply to thread
Message
[QUOTE="gildomar, post: 4621180, member: 428575"] [h2]Homework Statement [/h2] Prove the relationship between the momentum of the neutrino or nucleon in an elastic scattering of them in the center of mass frame is [itex]p'^{2}[/itex]=[itex]m_{1}E_{2}/2[/itex], where p' is the momentum of the neutrino or nucleon in the center of mass frame, [itex]m_{1}[/itex] is the mass of the nucleon, and [itex]E_{2}[/itex] is the relativistic energy of the neutrino in the laboratory frame. Assume the nucleon is initially at rest in the laboratory frame. [h2]Homework Equations[/h2] p'=[itex]\frac{1}{\sqrt{1-(v/c)^{2}}}[/itex][p-[itex]\frac{vE}{c^{2}}[/itex]] [itex]p'_{1}[/itex]: momentum of nucleon in center of mass frame [itex]p_{1}[/itex]: momentum of nucleon in laboratory frame [itex]p'_{2}[/itex]: momentum of neutrino in center of mass frame [itex]p_{2}[/itex]: momentum of neutrino in laboratory frame [itex]E'_{1}[/itex]: energy of nucleon in center of mass frame [itex]E_{1}[/itex]: energy of nucleon in laboratory frame [itex]E'_{2}[/itex]: energy of neutrino in center of mass frame [itex]E_{2}[/itex]: energy of neutrino in laboratory frame v: velocity of center of mass frame relative to laboratory frame [itex]E^{2}[/itex]=[itex]p^{2}[/itex][itex]c^{2}[/itex]+[itex]m^{2}_{0}[/itex][itex]c^{2}[/itex] [h2]The Attempt at a Solution[/h2] First find the velocity of the center of mass frame: [itex]p'_{1}[/itex] + [itex]p'_{2}[/itex]=0 [itex]\gamma[/itex][[itex]p_{1}[/itex]-[itex]\frac{vE_{1}}{c^{2}}[/itex]]+[itex]\gamma[/itex][[itex]p_{2}[/itex]-[itex]\frac{vE_{2}}{c^{2}}[/itex]]=0 [itex]p_{1}[/itex]-[itex]\frac{vE_{1}}{c^{2}}[/itex]+[itex]p_{2}[/itex]-[itex]\frac{vE_{2}}{c^{2}}[/itex]=0 Since the nucleon is at rest in the laboratory frame, [itex]p_{1}[/itex] is 0: [itex]p_{1}[/itex]-[itex]\frac{vE_{1}}{c^{2}}[/itex]+[itex]p_{2}[/itex]-[itex]\frac{vE_{2}}{c^{2}}[/itex]=0 [itex]0[/itex]-[itex]\frac{vE_{1}}{c^{2}}[/itex]+[itex]p_{2}[/itex]-[itex]\frac{vE_{2}}{c^{2}}[/itex]=0 [itex]p_{2}[/itex]=[itex]\frac{vE_{1}}{c^{2}}[/itex]+[itex]\frac{vE_{2}}{c^{2}}[/itex] [itex]p_{2}[/itex]=[itex]\frac{vE_1+vE_2}{c^2}[/itex] [itex]p_{2}[/itex]=[itex]\frac{v(E_1+E_2)}{c^2}[/itex] [itex]\frac{p_{2}c^2}{E_1+E_2}[/itex]=v Then get an expression for the squared momentum of the nucleon in the center of mass frame, using the above for the velocity, 0 for [itex]p_{1}[/itex], and [itex]m_{1}c^{2}[/itex] for [itex]E_{1}[/itex]: [itex]p'_{1}[/itex]=[itex]\frac{1}{\sqrt{1-(v/c)^{2}}}[/itex][[itex]p_{1}[/itex]-[itex]\frac{vE_{1}}{c^{2}}[/itex]] [itex]p'_{1}[/itex]=[itex]\frac{1}{\sqrt{1-(v/c)^{2}}}[/itex][[itex]0[/itex]-[itex]\frac{vm_{1}c^{2}}{c^{2}}[/itex]] [itex]p'_{1}[/itex]=[itex]\frac{1}{\sqrt{1-(v/c)^{2}}}[{-vm_{1}}[/itex]] [itex]p'^{2}_{1}[/itex]=[itex]\frac{m^{2}_{1}v^{2}}{1-(v/c)^{2}}[/itex] [itex]p'^{2}_{1}[/itex]=[itex]\frac{m^{2}_{1}}{1-(\frac{p_{2}c}{E_1+E_2})^{2}}[/itex][itex](\frac{p_{2}c^{2}}{E_1+E_2})^{2}[/itex] [itex]p'^{2}_{1}[/itex]=[itex]\frac{m^{2}_{1}(E_1+E_2)^{2}}{(E_1+E_2)^{2}-p^{2}_{2}c^2}[/itex][itex]\frac{p^{2}_{2}c^{4}}{(E_1+E_2)^{2}}[/itex] [itex]p'^{2}_{1}[/itex]=[itex]\frac{(m^{2}_{1}c^2)(p^{2}_{2}c^2)}{(E_1+E_2)^{2}-p^{2}_{2}c^2}[/itex] [itex]p'^{2}_{1}[/itex]=[itex]\frac{(m^{2}_{1}c^2)(p^{2}_{2}c^2)}{(m_{1}c^2+E_2)^{2}-p^{2}_{2}c^2}[/itex] [itex]p'^{2}_{1}[/itex]=[itex]\frac{(m^{2}_{1}c^2)(p^{2}_{2}c^2)}{m^{2}_{1}c^{4}+2m_{1}c^{2}E_{2}+E^{2}_{2}-p^{2}_{2}c^2}[/itex] Since the book was written before it was discovered that neutrinos have mass, the [itex]p_{2}c[/itex] was assumed to be equal to just [itex]E_{2}[/itex]: [itex]p'^{2}_{1}[/itex]=[itex]\frac{(m^{2}_{1}c^2)(E^{2}_{2})}{m^{2}_{1}c^{4}+2m_{1}c^{2}E_{2}+E^{2}_{2}-E^{2}_{2}}[/itex] [itex]p'^{2}_{1}[/itex]=[itex]\frac{m^{2}_{1}c^{2}E^{2}_{2}}{m^{2}_{1}c^{4}+2m_{1}c^{2}E_{2}}[/itex] [itex]p'^{2}_{1}[/itex]=[itex]\frac{m_{1}E^{2}_{2}}{m_{1}c^{2}+2E_{2}}[/itex] And then the only way to get the answer from there would be to assume that [itex]E_{2}[/itex] was much greater than [itex]m_{1}c^2[/itex], but I'm not sure if I'm allowed to assume that. Or did I screw up the calculations somewhere? [/QUOTE]
Insert quotes…
Post reply
Forums
Homework Help
Advanced Physics Homework Help
Proving momentum equation for neutrino/nucleon scattering
Back
Top