When linear momentum is conserved?
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Such a situation implies that the rate of change of the total momentum of a system does not change, meaning this quantity is constant, and proving the principle of the conservation of linear momentum: When there is no net external force acting on a system of particles the total momentum of the system is conserved.
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Let us consider two bodies of masses m1 and m2 moving in straight line in the same direction with initial velocities u1 and u2. They collide for a short time ∆t. After collision, they move with velocities v1 and v2.
From 2nd law of motion,
Force applied by A on B = Rate of change of momentum of B
FAB = (m2v2-m2u2)/∆t
Similarly,
Force applied by B on A = Rate of change of momentum of A
FBA = (m1v1-m1u1)/∆t
From Newton’s 3rd law of motion,
FAB = -FBA
Or, (m2v2-m2u2)/∆t = -(m1v1-m1u1)/∆t
Or, m2v2-m2u2 = -m1v1+m1u1
Or, m1u1 + m2u2 = m1v1 + m2v2
From 2nd law of motion,
Force applied by A on B = Rate of change of momentum of B
FAB = (m2v2-m2u2)/∆t
Similarly,
Force applied by B on A = Rate of change of momentum of A
FBA = (m1v1-m1u1)/∆t
From Newton’s 3rd law of motion,
FAB = -FBA
Or, (m2v2-m2u2)/∆t = -(m1v1-m1u1)/∆t
Or, m2v2-m2u2 = -m1v1+m1u1
Or, m1u1 + m2u2 = m1v1 + m2v2
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