Derivation of law of conservation of momentum
Answers
explanation
always conserved if no external force acts on an object or system of objects.
Consider a collision between two balls wherein there occurs no energy losses during the collision.
Momentum of the two balls before collision,
p1i = m1u1
p2i = m2u2
Total momentum of the system of two balls before collision, pi = p1i + p2i = m1u1 + m2u2.
During the collision m1 exerts an action force F12 on m2.
In response, from Newton"s third law, m2 exerts a reaction on m1, that is, F21, such that F12 = -F21
Negative sign implies that the two forces are directed in opposite directions.
After the collision they undergo change in velocity and the corresponding change in momentum. Momenta of the two balls after collision p1f = m1v1
p2f = m2v2
Total momentum of the system of two balls after collision, pf = p1f +p2f = m1v1 + m2v2
Also from Newton"s Second Law:
begin mathsize 12px style Force equals fraction numerator Change space in space momentum over denominator Time space interval end fraction
rightwards arrow straight F subscript 12 equals fraction numerator Change space in space momentum space produced space in space mass space straight m subscript 2 over denominator Collision space time end fraction
rightwards arrow straight F subscript 12 equals fraction numerator straight m subscript 2 straight v subscript 2 minus straight m subscript 2 straight u subscript 2 over denominator straight t end fraction
Similarly comma
rightwards arrow straight F subscript 21 equals fraction numerator straight m subscript 1 straight v subscript 1 minus straight m subscript 1 straight u subscript 1 over denominator straight t end fraction
From space Newton apostrophe straight s space third space law comma space straight F subscript 12 equals negative straight F subscript 21
rightwards arrow open parentheses fraction numerator straight m subscript 2 straight v subscript 2 minus straight m subscript 2 straight u subscript 2 over denominator straight t end fraction close parentheses equals negative open parentheses fraction numerator straight m subscript 1 straight v subscript 1 minus straight m subscript 1 straight u subscript 1 over denominator straight t end fraction close parentheses
rightwards arrow straight m subscript 1 straight u plus straight m subscript 2 straight u subscript 2 equals straight m subscript 1 straight v subscript 1 plus straight m subscript 2 straight v subscript 2 end style
Total momentum before collision = Total momentum after collision
Hence this equation implies that if no external force acts on the system of two colliding balls, the total momentum is conserved.
Answer:
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Explanation:
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