$(a)$ State the law of conservation of momentum.
$(b)$ Mention two factors which determine the momentum of a body.
$(c)$ Prove the law of conservation of momentum with clear explanation,diagram and equation.

(N/A) The law states that if no external force acts on a system of interacting bodies,the total momentum of the system remains constant.
$(b)$ The two factors that determine the momentum $(p)$ of a body are:
$1$. Mass $(m)$ of the body.
$2$. Velocity $(v)$ of the body.
$(c)$ Consider two objects $A$ and $B$ with masses $m_{1}$ and $m_{2}$ moving in the same direction with initial velocities $u_{1}$ and $u_{2}$ respectively,where $u_{1} > u_{2}$. They collide for a time $t$. During collision,$A$ exerts a force $F_{AB}$ on $B$,and $B$ exerts a force $F_{BA}$ on $A$.
Initial momentum of $A = m_{1}u_{1}$
Initial momentum of $B = m_{2}u_{2}$
Final momentum of $A = m_{1}v_{1}$
Final momentum of $B = m_{2}v_{2}$
According to Newton's second law,the force exerted by $A$ on $B$ is $F_{AB} = m_{2} \frac{(v_{2} - u_{2})}{t}$.
The force exerted by $B$ on $A$ is $F_{BA} = m_{1} \frac{(v_{1} - u_{1})}{t}$.
According to Newton's third law,$F_{AB} = -F_{BA}$.
$m_{2} \frac{(v_{2} - u_{2})}{t} = -m_{1} \frac{(v_{1} - u_{1})}{t}$
$m_{2}(v_{2} - u_{2}) = -m_{1}(v_{1} - u_{1})$
$m_{2}v_{2} - m_{2}u_{2} = -m_{1}v_{1} + m_{1}u_{1}$
$m_{1}u_{1} + m_{2}u_{2} = m_{1}v_{1} + m_{2}v_{2}$
This proves that the total initial momentum equals the total final momentum.