Conservation of Momentum
The law of conservation of momentum states that
"the total moment before the collision equals to total momentum after collision."
Derivation:
Lets us consider two object of masses m₁ and m₂ moving the velocity of u₁ and u₂ respectively. Then after the collision the velocity of the object changes to v₁ and v₂ respectively.
Before collision,
Object P:
mass = m₁
velocity = u₁
Object Q:
mass = m₂
velocity = u₂
Initial momentum of object P and Q is equal to m₁u₁ and m₂u₂ respectively.
After collision,
Object P:
mass = m₁
velocity = v₁
Object Q:
mass = m₂
velocity = v₂
Final momentum of object P and Q is equal to m₁v₁ and m₂v₂ respectively.
So, therefore
Rate of change of momentum of object P= F₂ = (m₁v₁ - m₁u₁) / t
Force exerted by object Q on P = (m₁v₁ - m₁u₁) / t
Rate of change of momentum of object Q = F₁ = (m₂v₂ - m₂u₂) / t
Force exerted by object P on Q = (m₂v₂ - m₂u₂) / t
According to Newtons Third Law of Motion
Every action has an equal and opposite reaction, the magnitude force exerted by object P is equal to the magnitude force exerted by object Q.
So, it is
F₂ = - F₁ ( negative sign is used since force acts on opposite direction )
(m₁v₁ - m₁u₁) / t = -[ (m₂v₂ - m₂u₂) / t]
after 't' gets cancelled on both side
m₁v₁ - m₁u₁ = -[ m₂v₂ - m₂u₂ ]
after removing bracket
m₁v₁ - m₁u₁ = - m₂v₂ + m₂u₂
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
