Question

State the law of conservation of momentum. Write its mathematical derivation.​

Answer 1

Answer:

The law of conservation of momentum states that for two objects colliding in an isolated system, the total momentum before and after the collision is equal. This is because the momentum lost by one object is equal to the momentum gained by the other.

Conservation of momentum is derived from Newton's laws of motion. Newton's third law states that every action has an equal but opposite reaction; the force that one object A exerts on object B is equal but opposite to the force that object B exerts on object A. By Newton's second law, this force is equal to the product of the mass and the acceleration of the objects, so the product of the mass and acceleration of object A is equal but opposite to the product between the mass and acceleration of object B.

Explanation:

Derivation of Conservation of Momentum

Let us consider a situation wherein: a truck of mass m1, velocity u1 and its momentum = m1u1 and a car of mass m2, velocity u2 and its momentum m2u2; are moving in the same direction but with different speeds. Therefore, total momentum=m1u1 + m2u2.

Now suppose the car and truck collide for a short time t, their velocities will change. So now the velocity of the truck and car become v1 and v2 respectively. However, their mass remains the same. Hence, now the total momentum = m1v1 + m2v2

Acceleration of car (a) = (v2–u2)/t

Also, F = ma

F1  = Force exerted by truck on the car

F1 = m2(v2–u2)/t

Acceleration of truck =(v1–u1)/t

F2  =   m1(v1–u1)/t and F1 = –F2

m2(v2– u2)/t = –m1(v1– u1)/t

m2v2–m2u2 = –m1v1+m1u1

or m1u1+m2u2  = m2v2+m1v1

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Answer 2

Newton's third law states that for a force applied by an object A on object B, object B exerts back an equal force in magnitude, but opposite in direction. This idea was used by Newton to derive the law of conservation of momentum. ... B=m_{2}(v_{2}-u_{2}) (change in momentum of particle B)