Question

State and prove the law of conservation of linear momentum.

Answer 1

Law of conservation of momentum : Law of conservation of momentum states, in the absence of a net external force on the system, the momentum of the system remains unchanged. 

Explanation :

• Let two marbles with masses m₁ and m₂ travel with different velocities u₁ and u₂ in the same direction along a straight line. 
• If u₁ > u₂, they collide each other and the collision lasts for time ’t’.
• During collision, each marble exerts force on other marble. [F₁₂ and F₂₁ ] 
• Let v₁ and v₂ be the velocities of the marbles after collision.

Now look at the table.

	Marble 1	Marble 2
Momentum before collision	m1u1	m2u2
Momentum after collision	m1v1	m2v2
change in momentum (∆p)	m1v1 - m1u1	m2v2 - m2u2
Rate of change of momentum \(\frac{\Delta p}{\Delta t}\)	\(\frac{(m_1v_1\, -\,m_1u_1)}{t}\)	\(\frac{(m_2v_2\, -\,m_2u_2)}{t}\)

According to newton's third law,

F₁₂ = -F₂₁  ⇒ \(\frac{\Delta p_1}{t}=\frac{\Delta p_2}{t}\)

\(\Rightarrow\frac{m_1v_1\,-\,m_1u_1}{t}=\frac{-(m_2v_2\,-\,m_2u_2)}{t}\)

⇒ m₁v₁ - m₁u₁= -m₂v₂ + m₂u₂

⇒ m₁v₁ + m₂v₂ = m₁u₁ + m₂u₂ 

  \(\therefore\) m₁u₁ + m₂u₂ = m₁v₁ +m₂v₂

• The total momentum before collision is equal to total momentum after collision. 
• Hence, the total momentum remains unchanged before and after collision.