Derive the law of conservation of linear momentum from the newton's 3rd law.
Answers
When two bodies collide, the forces that one acts on the other constitute the “action-reaction” pair. Each body receives an equal and opposite impulse F Δt. As such, the change in linear momentum of each body is equal and opposite. The resultant change in linear momentum of the system is zero. In other words, the total momentum of the system of two colliding bodies remains constant.
The principle of conservation of momentum states that the total momentum of a system of colliding objects remains constant provided no resultant external force acts on the system.Explanation of Conservation Of Linear Momentum Using Newton’s Laws
Consider an isolated system of two objects m1 and m2 whereby m1 is striking m2. (An isolated system is one which has no external forces acting on it.)
Mathematically, we can deduce the principle of conservation of linear momentum by applying Newton’s second and third laws to the colliding bodies as illustrated below:
From Newton’s third law, the contact forces of impact are action and reaction pair
F21 = – F12
If the two bodies are in contact for a time of Δt,
F21 Δt = – F12 Δt ——(1)
impulse on m1 = F21 Δt (Recall that impulse = change in momentum)
F12 Δt = m1v1 – m1u1
impulse on m2 = F12?t
F21 Δt = m2v2 – m2u2
Substitute into (1),
(m1v1– m1u1) = – (m2v2 – m2u2)
Rearranging => m1u1 + m2u2 = m1v1 + m2v2
Thus, total linear momentum before collision = total linear momentum after collision,
which proves the principle of conservation of linear momentum.
Note:
• F12 and F21 are internal forces of the system. i.e. forces which arise between the interacting objects of the system.
• If there is an external force present, for example, friction, then the law of conservation of linear momentum is NOT obeyed.
• The angular momentum of the system is ALSO obeyed, but is beyond the H2 syllabus.
Worked Example on Conservation of momentum
Calculate the recoil velocity of a 4.0 kg rifle which shoots a 0.050 kg bullet at a speed of 280 m s-1.
Solution:
Newton's second law of motion says that rate of change of linear momentum of a body is equal to the net external force applied on it. If the net external force acting on a body is zero, then the rate of change of momentum is also zero, which means that there is no change in momentum.