State and prove the principles of conservation of linear momentum give examples
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Answer :
According to the law of conservation of momentum, when two or more bodies act upon each other their total momentum remains constant provided no external forces are acting. So, momentum is never created or destroyed.
When this law is applied for a collision between two bodies, the total momentum of the colliding bodies before the collision is equal to the total momentum after the collision.
Consider two particles say A and B of mass m 1 and m2 collide with each other and forces acting on these particles are only the ones they exert on each other. There is no external force in play.
Let u1 and v1 be the initial and final velocities of particle A and similarly, u2 and v2 for particle B. Let the two particles be in contact for a time t.
So, Change in momentum of A= m 1 (v 1 −u 1 ). ..... eq. 1
Change in the momentum of B=m 2 (v 2 −u 2 ). ..... eq. 2
During the collision, let A impart an average force equal to F BA on B and let B exert an average F AB on A.
We know that from third law of motion F BA =−F AB ..... eq. 3
We know that, force equals change in momentum per unit time, therefore: F BA =m 2 ×a 2 = m2(v2−u2) / t
FAB=m1×a1= m1(v1−u1) / t
Putting above two in equation 3 we get,
m2(v2−u2) / t = m1(v1−u1) / t ... eq. 4
Canceling t on both sides and rearranging the equation we get
m 1 u 1 +m 2 u 2 =m 1 v 1 +m 2 v 2 ...... eq. (5)
Now, m 1 u 1 +m 2 u 2 represents the total momentum of particles A and B before collision and m 1 v 1 +m 2 v 2 represents the total momentum of particles after the collision.
This means that, total momentum before collision = total momentum after the collision Equation 5 which states,
m 1 u 1 +m 2 u 2 =m 1 v 1 +m 1 v 2 ,
is known as the law of conservation of momentum.