Question

A particle of mass M, initially at rest, decays into two particles with rest masses m1 and m2 respectively. Show that the total energy of the mass m1 is: E1 = c2[ M2+m12-m22 ] / 2M

Answer 1

Answer:

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Explanation:

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Answer 2

Answer:

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Explanation:

Explanation:

Total energy:

E = E₁ + E₂   ( 1 )

such that: E = m₀c²

Conservartion of linear momentum:

P = p₁ + p₂  

such that P = 0 ⇒ p₁ = - p₂

then:

|p₁|² = |p₂|²    ( 2 )

Relation energy-momentum:

E² = m²c⁴ + p²c²

For particle 1:

E₁² = m₁²c⁴ + p₁²c²    ( 3 )

For particle 2:

E₂² = m₂²c⁴ + p₂²c²   ( 4 )

Eq. ( 3 ) and ( 4 ) in ( 2 ):

E₁/c² - m₁²c² = E₂/c² - m₂c²

if: E₂ = E - E₁ then

E₁/c² - m₁c² = (E - E₁)²/c² - m₂c² ⇒

- m₁c² = E²/c² - 2E₁E/c² - m₂c² ⇒

E₁ = E/c² + (m₂ - m₂)c⁴/2E ⇒

E₁ = [E₂ + (m₂ - m₂)c⁴]/2E ⇒

if E = m₀c², then

E₁ = [m₀² + (m₂ - m₂)c²]/2m₀

QED.