Question

what is conservation of momemtum

Answer 1

◆ CONSERVATION OF MOMENTUM ◆

According to the law of conservation of momentum the total momentum of a system (or an object) remains constant if no net external unbalanced force acts on the system.

That is, momentum of a system , p = constant, if net external force acting on it is zero (i.e. F = 0).

◆ PROOF ◆

Consider a system consisting of two bodies A and B of masses $m_1$ and $m_2$ respectively.

Let these bodies are moving with velocities $u_1$ and $u_2$.

Let $u_1$ > $u_2$. Let these bodies collide with each other for a small interval of time $dt$.

At the time of collision, body A exerts a force F on body B and body B exerts equal and opposite force (-F) on body A.
Due to these forces, the momentum of the bodies changes.

Let $v_1$ and $v_2$ be the victories of body A and body B respectively after the collision.

Initial momentum of body A = $m_1 u_1$
Initial momentum of body B = $m_2 u_2$
Final momentum of body A = $m_1v_1$
Final momentum of body B = $m_2v_2$

Total momentum of the system before collision = $m_1u_1$ + $m_2u_2$
Total momentum of the system after collision = $m_1v_1$ + $m_2v_2$

Now, the rate of change of momentum of body A = change in momentum of body A/ Time Taken

= Final momentum - Initial momentum / Time Taken

= $\frac{ (m_1 v_1- m_1 u_1) }{dt}$

According to Newton's second law of motion,
Force acting on body A = rate of change of momentum of body A

i.e. , $F = \frac{ (m_1 v_1- m_1 u_1) }{dt} ..........Equ (1)$

Now, rate of change of momentum of body B = change in momentum of body B/Time Taken

= $\frac{ (m_2v_2- m_2 u_2) }{dt}$

According to Newton's second law of motion, Force acting on body B = rate of change of momentum of body B

i.e. $-F = \frac{ (m_2v_2- m_2 u_2) }{dt}$

Or , $F = \frac{-( m_2 v_2- m_2 u_2)}{dt} = \frac{-( m_2 v_2- m_2 u_2)}{dt}..........Equ(2)$ Equating eqns. (1) and (2) , we get $\frac{-( m_2 v_2- m_2 u_2)}{dt} = \frac{-( m_1v_1- m_1u_1)}{dt}$ Or, $m_1 u_1 + m_2 u_2= m_1 v_1+ m_2 v_2$ i.e. Total momentum of the system ( i.e. bodies A and B ) before collision = Total momentum of the system ( i.e. bodies A and B ) After collision. This is the law of conservation of linear momentum.