Question

write the law of conservation of momentum and prove it.​

Answer 1

• $\mathfrak{\underline{\underline{Law\:of\:conservation\:of\:momentum:-}}}$

The total momentum of a system of two or more objects before & after collision remains the same / constant. Provided no external force acting on it.

=> Initial momentum = Final momentum

• $\mathfrak{\underline{\underline{Proof:-}}}$

Suppose, two objects A and B of masses $\sf{M}_{1}$ and $\sf{M}_{2}$ are moving with velocities $\sf{U}_{1}$ and $\sf{U}_{2}$ collide with each other such that their velocities changes to $\sf{V}_{1}$ and $\sf{V}_{2}$ respectively.

★ $\textsf{\underline{\underline{Before\:Collision:-}}}$

Initial momentum of object A = $\sf{M}_{1}{U}_{1}$.

Initial momentum of object B = $\sf{M}_{2}{U}_{2}$

★ $\textsf{\underline{\underline{After\:collision:-}}}$

Final momentum of object A = $\sf{M}_{1}{U}_{1}$.

Final momentum of object B = $\sf{M}_{2}{U}_{2}$

•°• Rate of change in Momentum in A

$\sf{F}_{1}$ = $\sf{m}_{1}$${(V}_{1}-{U}_{1)}$/t --------1.

•°• Rate of change in Momentum in B

$\sf{F}_{2}$ = $\sf{m}_{2}$${(V}_{2}-{U}_{2)}$/t ----------2.

We know from third law of motion,

$\sf{F}_{1}$ = $\sf{-F}_{2}$

So, $\sf{m}_{1}$${(V}_{1}-{U}_{1)}$/t = $\sf{-m}_{2}$${(V}_{2}-{U}_{2)}$/t

=$\sf{M}_{1}{V}_{1-}$$\sf{M}_{2}{U}_{2}$=$\sf{-M}_{2}{V}_{2}$+$\sf{M}_{1}{U}_{1}$

Thus, Initial momentum = Final momentum,

Here $\sf{(M}_{1}{U}_{1}+{M}_{2}{M}_{2)}$ is the total momentum of two objects A & B before collision.

And $\sf{(M}_{1}{U}_{1}+{M}_{2}{M}_{2)}$ is the total momentum of two objects A and B after collision.