Question

Derive the mathematical expression for law of conservation of momentum.

Answer 1

Law:

For a collision occurring between object 1 and object 2 in an isolated system, the total momentum of the two objects before the collision is equal to the total momentum of the two objects after the collision.

Derivation:

Consider two colliding particles A and B whose masses are m1 and m2 with initial and final velocities as u1 and v1 of A and u2 and v2 of B. The time of contact between two particles is given as t.

A=m1(v1−u1) (change in momentum of particle A)

B=m2(v2−u2) (change in momentum of particle B)

F(BA)=−F(AB) (from third law of motion)

F(BA)=m2×a2=m2(v2−u2)t

FAB=m1×a1=m1(v1−u1)t

m2(v2−u2)t=−m1(v1−u1)t

m1u1+m2u2=m1v1+m2v2

Inference:

Therefore, above is the equation of law of conservation of momentum where, m1u1+m2u2 is the representation of total momentum of particles A and B before collision and m1v1+m2v2 is the representation of total momentum of particles A and B after collision.

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)