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# State and derive law of conservation of momentum.

**Law of conservation of momentum** states that-

For two or more bodies in an isolated system acting upon each other, their total momentum remains constant unless an external force is applied.

Therefore, momentum can neither be created nor destroyed.

Derivation of the conservation of momentum:

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.

**$\displaystyle A=m_{1}( v_{1} -u_{1})$ **(change in momentum of particle A)

**$\displaystyle B=m_{2}( v_{2} -u_{2})$ **(change in momentum of particle B)

**$\displaystyle F_{BA} =-F_{AB}$ **(from third law of motion)

**$\displaystyle F_{BA} =m_{2} \times a_{2} \ $**

** $\displaystyle =$$\displaystyle \frac{m_{2}( v_{2} -u_{2})}{t}$**

**
**

**$\displaystyle F_{AB} =m_{1} \times a_{1} \ $**

** $\displaystyle =m_{1}( v_{1} -u_{1}) \ m_{2}( v_{2} -u_{2}) \ t$**

** $\displaystyle =-m_{1}( v_{1} -u_{1}) t\ m_{1} u_{1} +m_{2} u_{2}$**

** $\displaystyle =-m_{1} v_{1} +m_{2} v_{2}$**

**
**

Therefore, above is the equation of law of conservation of momentum where, **$\displaystyle =-m_{1} u_{1} +m_{2} u_{2}$** is the representation of total momentum of particles A and B before the collision and **$\displaystyle =-m_{1} v_{1} +m_{2} v_{2}
$ **is the representation of total momentum of particles A and B after the collision.