First off momentum = mass x velocity. You can prove the conservation of momentum (i.e. that it stays the same unless a force acts upon a body) from Newton's laws.
Force = mass x acceleration
Acceleration is the change of velocity over time so:
Force = mass x ((velocity1 - velocity2) / time)
Force = ((mass x velocity1) - (mass x velocity2)) / time
Because mass x velocity is momentum, we can say:
Force = (momentum1 - momentum2) / time
Force x time = momentum1 - momentum2
Meaning a change in momentum is caused by force multiplied by time. If there is no force (or no time for it to be applied), the left hand side of the equation is zero, meaning momentum1 = momentum2 (and therefore there is no change in momentum).
Therefore, without a force being applied for a period of time, momentum is conserved.
Consider 2 objects A and B of masses ma and mb .Let they are traveling with the initial velocities of ua and ub. Let ua is greater than ub. Let them collide and collision lasts of time t. Let the final velocities be va and vb. Let there is no external unbalanced force applied on them.
By 3rd law of motion
FAB = -FBA (1)
By 2nd law of motion
F = ma. (2)
By 1st equation of motion
a = v-u/t. (3)
Using (2) and (3) in (1)
FAB = ma(va-ua/t). (4)
FBA = mb(vb-ub/t). (5)
Using (4) and (5) in (1)
ma(va-ua/t) = -(mb(vb-ub/t))
t will get cancelled
(mv)a - (mu)a = -(mb)b + (mu)b
maua + mbub = mava + mbvb