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

[Guaicai]

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 ?

 

[Simon Bridge]

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.

 

[Guaicai]

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 )

 

[haruspex]

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?