State and prove law of conservation of momentum.

Solution

Law of conservation of momentum states that total momentum of system remains conserved in the absence of external force.
Proof:
Consider a body of mass m₁ moving with veocity $straight u with rightwards harpoon with barb upwards on top subscript 1$ striking against another body of mass m₂ moving with velocity $straight u with rightwards harpoon with barb upwards on top subscript 2$ .
Let, the two bodies remain in contact with each other for a small interval $increment straight t$ .
Let, $straight F with rightwards harpoon with barb upwards on top subscript 12 space$ be the average force exerted by mass m₁ on m₂ and let $straight F with rightwards harpoon with barb upwards on top subscript 21$ be the force on m₂ due to m_1.Let, v₁ and v₂ be the velcoities of two bodies after collision.

Momentum of mass m₁ before collision = m₁u₁
Momentum of mass m₂ after collision = m_{2 $stack straight u subscript 2 with rightwards harpoon with barb upwards on top$}Momentum of mass m₁ after collision = m₁v₁
By using the defintion of impulse, change in momentum of mass m₁ is,
$straight F with rightwards harpoon with barb upwards on top subscript 12 increment t space equals space m subscript 1 stack v subscript 1 with rightwards harpoon with barb upwards on top space minus space m subscript 1 u with rightwards harpoon with barb upwards on top subscript 1 space space space space space space space space space space... left parenthesis 1 right parenthesis space C h a n g e space i n space m o m e n t u m space o f space m a s s space m subscript 2 subscript space end subscript space i s comma space straight F with rightwards harpoon with barb upwards on top subscript 21 increment t space equals space space m subscript 2 stack v subscript 2 with rightwards harpoon with barb upwards on top space minus space m subscript 2 u with rightwards harpoon with barb upwards on top subscript 2 space space space space end subscript space space space space space... left parenthesis 2 right parenthesis space A d d i n g space e q u a t i o n s space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis comma space w e space h a v e space space space space space space open parentheses straight F with rightwards harpoon with barb upwards on top subscript 12 increment t space plus straight F with rightwards harpoon with barb upwards on top subscript 21 increment t space close parentheses space equals space left parenthesis m subscript 1 stack v subscript 1 with rightwards harpoon with barb upwards on top space minus space m subscript 1 u with rightwards harpoon with barb upwards on top subscript 1 right parenthesis space plus space left parenthesis space m subscript 2 stack v subscript 2 with rightwards harpoon with barb upwards on top space minus space m subscript 2 u with rightwards harpoon with barb upwards on top subscript 2 space space space space end subscript right parenthesis rightwards double arrow space left parenthesis straight F with rightwards harpoon with barb upwards on top subscript 12 plus straight F with rightwards harpoon with barb upwards on top subscript 21 right parenthesis increment t space space equals space left parenthesis m subscript 1 stack v subscript 1 with rightwards harpoon with barb upwards on top plus m subscript 2 stack v subscript 2 with rightwards harpoon with barb upwards on top right parenthesis space minus space left parenthesis space m subscript 1 u with rightwards harpoon with barb upwards on top subscript 1 space minus m subscript 2 u with rightwards harpoon with barb upwards on top subscript 2 space space space space end subscript right parenthesis S i n c e comma space F with rightwards harpoon with barb upwards on top space subscript 12 space a n d space F with rightwards harpoon with barb upwards on top subscript 21 space a r e space e q u a l space a n d space o p p o s i t e comma space space space space space space space space space space space space space space space space space space space space space space space space space space space space F with rightwards harpoon with barb upwards on top space subscript 12 space plus F with rightwards harpoon with barb upwards on top subscript 21 space equals space 0 T h u s comma space left parenthesis m subscript 1 stack v subscript 1 with rightwards harpoon with barb upwards on top plus m subscript 2 stack v subscript 2 with rightwards harpoon with barb upwards on top right parenthesis space equals space left parenthesis m subscript 1 u with rightwards harpoon with barb upwards on top subscript 1 space minus m subscript 2 u with rightwards harpoon with barb upwards on top subscript 2 space space space space end subscript right parenthesis space M o m e n t u m space a f t e r space c o l l i s i o n equals space M o m e n t u m space b e f o r e space c o l l i s i o n H e n c e comma space m o m e n t u m space o f space i s o l a t e d space s y s t e m space i s space c o n s e r v e d. space$

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Laws Of Motion

State and prove law of conservation of momentum.

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