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

AI Thread Summary
In a two-particle collision, the invariant mass squared (M²) is expressed as M² = (E1 + E2)² - (p1 + p2)², where E represents energy and p represents momentum. The equation incorporates the masses of the particles and their energies, with E being defined as mc², using c = 1 for simplification. A key point of discussion is the dot product of momentum vectors, specifically how to derive the term p1 ⋅ p2 from the expansion of (p1 + p2)². The algebraic expansion reveals that (p1 + p2)² includes terms for the individual momenta and their dot product. Understanding this relationship is crucial for analyzing energy and mass in particle collisions.
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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Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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