Question

prove mathematically that initial momentum of a system of two colliding masses is equal to their final momentum

Answer 1

Answer is yes it is equal due to their same masses and with their same velocities but after colliding their masses remain same but velocities change

Answer 2

Answer: The initial momentum equal to the final momentum.

Explanation:

Let us considered two object A and B having masses of $m_{A}$ and $m_{B}$ and initial velocities $u_{A}$, $u_{B}$  and final velocities $v_{A}$, $v_{B}$before and after collision.

The change in momentum before collision

$m_{A}u_{A}+m_{B}u_{B}$.....(I)

The change in momentum after collision

$m_{A}v_{A}+m_{B}v_{B}$.....(II)

We know that,

The force is the product of mass and acceleration.

The acceleration is the rate of change of velocity of the object .

The force $F_{AB}$ is

$F _{AB}= \dfrac{m_{B}v_{B}-m_{B}u_{B}}{t}$

The force $F_{BA}$ is

$F _{BA}= \dfrac{m_{A}v_{A}-m_{A}u_{A}}{t}$

Using newton's third law

$F_{AB}=-F_{BA}$

$\dfrac{m_{B}v_{B}-m_{B}u_{B}}{t}=-(\dfrac{m_{A}v_{A}-m_{A}u_{A}}{t})$

$m_{B}v_{B}-m_{B}u_{B}=-m_{A}v_{A}+m_{A}u_{A}$

$m_{A}u_{A}+m_{B}u_{B}=m_{A}v_{A}+m_{B}v_{B}$

The initial momentum of a system is equal to the final momentum after collision.

Hence, The initial momentum equal to the final momentum.