What is the relationship between energy and mass in a two-particle collision?

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Homework Help Overview

The discussion revolves around the relationship between energy and mass in the context of a two-particle collision, specifically focusing on the invariant mass and the role of momentum in the equations governing the system.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants are examining the equation for invariant mass and questioning how the dot product of momentum vectors arises in the context of the squared momentum sum. There are attempts to derive the relationship between the squared momentum and the dot product, with some participants expressing confusion over extra terms that appear during their algebraic manipulations.

Discussion Status

The discussion is ongoing, with participants actively engaging in algebraic exploration and questioning the derivation process. Some guidance has been offered in the form of hints to expand brackets, but no consensus or resolution has been reached regarding the derivation of the dot product from the squared momentum sum.

Contextual Notes

Participants are working under the assumption that the speed of light is set to one, which may influence their calculations and interpretations of energy and mass relationships.

Guaicai
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Homework Statement


In a two-particle collision the square of the invariant mass is.
M is total mass of the system (M2 is the square of the total mass)
m is the mass of each particles
E is the energy of each particles
p is momentum vector of eache particles

Homework Equations


M2 = (E1+E2)2 - (p1+p2)2 = m12+m22+2(E1E2 - p1 ⋅ p2)

now know the each energy E = mc2, here set the c =1 (Speed of Light),

The Attempt at a Solution


But how the momentum have dot product in this equation ? How the dot product p1 ⋅ p2 can derive from (p1+p2)2 ?
 
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Guaicai said:
But how the momentum have dot product in this equation ? How the dot product p1 ⋅ p2 can derive from (p1+p2)2 ?
Try doing the algebra... hint: expand the brackets.
 
Simon Bridge said:
Try doing the algebra... hint: expand the brackets.
Yeah , i was tried to deriving (p1+p2)2 to the single dot product as above ,but always have the extra term.
(p1+p2)2
=p12+p22+2p1p2
=p12+p22+( p1 ⋅ p2 )
 
Guaicai said:
Yeah , i was tried to deriving (p1+p2)2 to the single dot product as above ,but always have the extra term.
(p1+p2)2
=p12+p22+2p1p2
=p12+p22+( p1 ⋅ p2 )
What is the relationship between E1 and m1?
 
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