Name: Relative Motion and Frame of Reference
Uploaded: 2019-07-19
Duration: 6 min 27 s
Description: Are you moving right now? It depends upon which frame you are in and from which frame you are observing. Explore relative motion and reference frames.

Question

Write the law of conservation of momentum. Derive from it Newton's third law of motion.

Accepted Answer

The law of conservation of momentum: The total momentum of an isolated system of interacting particles is conserved. Thus, in an isolated system (i.e. a system with no external force), mutual forces between pairs of particles in the system change the momentum in individual particles, but since the mutual forces for each pair are equal and opposite, the momentum changes cancel in pairs and the total momentum remains unchanged.

Take two colliding bodies A and B. Let the initial momentum of the two bodies be ${\vec{p}}_{A}$ and ${\vec{p}}_{B}$ . Let after colliding for a time interval of $∆ t$ their momentum are ${\vec{p}}_{A}^{'}$ and ${\vec{p}}_{B}^{'}$ . During the collision, the force applied by A on B is ${\vec{F}}_{B A}$ and force applied on A by B is ${\vec{F}}_{A B}$ . Applying the law of conservation of momentum for interacting bodies:

${\vec{p}}_{A} + {\vec{p}}_{B} = {\vec{p}}_{A}^{'} + {\vec{p}}_{B}^{'}$ .........(i)

By the second law of motion:

${\vec{p}}_{A}^{'} - {\vec{p}}_{A} = {\vec{F}}_{A B} ∆ t$ .......(ii)

${\vec{p}}_{B}^{'} - {\vec{p}}_{B} = {\vec{F}}_{B A} ∆ t$ .......(iii)

Using (i), (ii) and (iii) we get:

${\vec{F}}_{A B} = - {\vec{F}}_{B A}$ .........(iv)

(iv) shows that A and B apply equal and opposite forces on each other. Hence, every action has an equal and opposite reaction, which is Newton's third law of motion.

Write the law of conservation of momentum. Derive from it Newton's third law of motion.

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