In nature conservation laws play a very important role . The dynamics of motion of bodies can be analysed very effectively using conservation laws . There are three conservation laws conservation of angular momentum .By combining newton.s second and third laws we can derive the law of condervation of total linear momentum .
When two particles interact with each other , they equal and opposite forces on each other . The particles 1 exerts force ` vec(F_(12))` on particle 2 and particle 2 exerts an exactly equal and opposite force ` vec(F_(12))` on particle 1 according to Newton.s third law .
` vecF_(12) =- vec_(12)`
In terms of momentum of particles the force on each particle ( Newton.s second law ) can be written as
` vecF_(12) = ( d vec(P_1))/(d t ) and vecF_(21) = ( d vec(P_2))/(d t)`
here ` barP_1 ` is the momentum of particle 1 which changes due to the `vecF_(12)` exerted by particle 2 .Further `barP_(2)` is the momentum of particle 2. this changes due to `F_(12)` exerted by particle 1.
SUbstiute equation (2) in equation (1)
` ( d barP_1)/(dt ) = ( d vecP_2)/(d t)`
` ( d vecp _1)/( d t) + ( d barP_2)/( d t) =0`
` (d) /( dt) + ( vecp_1 + vecP_2) =0`
It implies that ` vecP_1 + vecP_2` = constacnt vector ( always ) .
` vecP_1 + vecP_2` is the total linear momentum of the two particles `( vecP_(n e t ) = vecP_1 + vecP_2)`. It is also called as total linear momentum of the system . here the two particles constitute the system . from this result the law of conservation of linear momentum can be stated as follows .
If there are no external forces acting on the system , then the system `( vecp_(n e t))` is always a constant vector .
THe forces ` vecF_(12)` are called the internal forces of the system because they act only between the two particle there is no external force acting on the twpo particles from outside . in such a case the total linear momentum of the system is a constatn vector or is conserved .
to find the recoil velocity of a gun when a bullet is fired from it .
Consider the firing of a gun , here the system is Gun + bullet .Initially the gun and bulllet are at rest , hence the total linear momentum of the system is zero . Let ` vecP_1` be the momentum of the bullet and ` vecP_2` the momentum of the before firing .Since initially both are at rest
total momentum before firing the gun zero ` vecP_1 + vecP_2 = 0`
According to the law of condervation of linear momentum , total momentum has to be zero after the firing also
when the gun is fired is exerted by the gun on the bullet in forward direction . Now the momentum of hte bullet from ` vecP_1 + vecP_1` to conserve the total linear momentum of the system the momentum of the gun must also change from ` vecP_2 ` to ` vecP_2`, due to the conservation of liinear momentum `, vecP_1 + vecP_2 =0` .It implies that ` vecP_1 =- vecP_2` the momentum of the gun is ecactly equal , but in the opposite direaction to the momentum of the bullet this is the reason after firing the gun suddenly moves backward with the meomentum `( - vecP_2)` . it is called . recoil momentum . . Th is is an example of conservation of total linear momentum .
