Question

Explain Conservation of linear momentum ? proof also ?

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Answer 1

Answer:

Conservation of linear momentum, general law of physics according to which the quantity called momentum that characterizes motion never changes in an isolated collection of objects; that is, the total momentum of a system remains constant.

Answer 2

${\underline{\boxed{\frak{\pmb{\quad★ \:Conservation \: of \: Linear \: Momentum :-\quad}}}}}$

→ It state that in an isolated system. The net Momentum of the whole system always remain constant is called change of linear momentum.

→ Let us consider two balls having Mass ${\sf{ M_{(1)} }}$ and ${\sf{ M_{(2)} }}$ Moving with some initial velocity ${\sf{ U_{(1)} }}$ and ${\sf{ U_{(2)} }}$ Respectively.

→ Now, After collision they acquires velocities ${\sf{ V_{(1)} }}$ and ${\sf{ V_{(2)} }}$

→ Let us assume that, the system is isolated i.e, there is no external force acting on it.

${\underline{\boxed{ \purple{\frak{\pmb{\quad★ Before \: \: Collision:-\quad}}}}} }$

$\implies{ \sf{We \: know, p = mv}}$

$\implies{\sf{ p _{(i1) \: } = m_{(1) \: }u _{(1) \: }}}$

$\implies{\sf{ p _{(i2) \: } = m_{(2) \: }u _{(2) \: }}}$

$\implies{\sf{ p _{(Ti) \: } = m_{(1) \: }u _{(1) \: } +m_{(2) \: }u _{(2) \: } }}$

${\underline{\boxed{ \red{\frak{\pmb{\quad★ \: After \: \: Collision:-\quad}}}}} }$

$\implies{\sf{ p _{(1f) \: } = m_{(1) \: }v _{(1) \: }}}$

$\implies{\sf{ p _{(2f) \: } = m_{(2) \: }v _{(2) \: }}}$

$\implies{\sf{ p _{(Tf) \: } = m_{(1) \: }v_{(1) \: } +m_{(2) \: }v_{(2) \: } }}$

$\implies{\sf{ p _{(Ti) \: } = \: p_{(Tf) }}}$

${\sf{ \implies{ m_{(1) \: }u _{(1) \: } +m_{(2) \: }u _{(2) \: } =m_{(1) \: }v _{(1) \: } +m_{(2) \: }v _{(2) }}}}$

${\underline{\boxed{ \green{\frak{\pmb{\quad★ \: Proof :- During \: \: Collision:\quad}}}}} }$

$\implies{\sf{F _{(21) }}}$ it's mean, Force acting on 2 due to 1.

$\implies{\sf{F _{(12) }}}$ it's mean, Force acting on 1 due to 2.

$\implies{ \sf{We \: know, f = ma \: [ Ⅱ law ]}}$

$\implies{\sf{F _{(21) } ={m _{(1) }{a_{(1) } }}}}$

$\implies{\sf{F _{(12) } ={m _{(2) }{a_{(2) } }}}}$

${ \sf{Using \: Ⅱ \: law : - }}$

$\implies{\sf{F _{(21) } = { - F _{(12) } }}}$

$\implies{\sf{ m _{(1) }{a_{(1) } ={ - m _{(2) }{a_{(2) } }}}}}$

Let, Time of Collisation is 't'

$\sf {\implies{ m _{(1) } = \frac{({ v _{(1) } -{ u _{(1) })}}}{t} }= { - m _{(2) } = \frac{({ v _{(2) } -{ u _{(2) })}}}{t} }}$

$\sf{ \implies\frac{1}{t}[ m_{(1)} \: (v_{(1)}u_{(1)}})] = \frac{ - 1}{t}[ m_{(2)} \: (v_{(2)}u_{(2)}) ]$

${ \sf{ \implies{m_{(1)}(v_{(1)} - u_{(1)} )= \cancel\frac {t}{t} \:[m_{(2)}(v_{(2)} -u_{(2)})]}}}$

${ \sf{ \implies{m_{(1)}(v_{(1)} - u_{(1)} )= - m_{(2)}(v_{(2)} - u_{(2)})}}}$

$\sf{ \implies{m_{(1)}v_{(1)} = {m_{(1)}u_{(1)} = - m_{(2)}v_{(2)} + {m_{(2)}u_{(2)} }}}}$

$\sf{ \implies{m_{(1)}v_{(1)} + {m_{(2)}v_{(2)} = m_{(1)}u_{(1)} - {m_{(2)}u_{(2)} }}}}$

$\implies{ \pink{ \boxed{ \sf{p_{(Tf)} = {p_{(Tf)}}}}}} \: { \sf{Hence, \: Proved.}}$

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