Question

Why is momentum conserved?

Answer 1

Step 1: Concept

1. The momentum of a system is always conserved because of the third law of motion.
2. Newton's Third Law: When two bodies are in interaction, they apply forces on each other that are equal in magnitude and opposite in direction.

Step 2: Explanation

1. We can explain the conservation of momentum by a simple collision of two objects in which the total momentum before the collision will be equal to the total momentum after the collision.
2. Let us consider a collision between two objects $A$ and$B$ when the two objects collide they apply force on each other. force on $A$ due to $B$ is $F_{AB}$ and due to Newton's third law, there is an equal and opposite force on $B$ due to$A$. $F_{BA}$

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

The forces act on objects when they are in contact. Since the time of contact is equal for both of the objects.

so, $t_{AB}=t_{BA}$

Hence the impulse on both the objects is equal and opposite in direction.

$F_{AB}.t_{A{B}_{}}=-F_{AB}.t_{A{B}_{}}$

We know that impulse is equivalent to the change in momentum. From this, we can say that change in momentum is equal but opposite in direction.

From this we can say that sum of the change in momentum is zero, which can be expressed as:

$m_A.∆v_{A_{}}=-m_B.∆v_B$

$m_A.∆v_{A_{}}+m_B.∆v_B=0$ Where, $∆v_A=v_{A_{f_{}}-}{v}_{A_{i_{}}}$

$∆v_B=v_{{B_{f_{}}}_{}}-v_{{B_{i_{}}}_{}}$

$m_A\left(v_{A_f-}{v}_{A_{i_{}}}\right)+m_B\left(v_{B_f}-v_{{B_i}_{}}\right)=0$

$m_A.v_{A_i}+m_{B_{}}.v_{Bi}=m_A.v_{A_f}+m_{B_{}}.v_{B_f}$ (Since we know that mass x velocity = momentum)

Total initial momentum = total final momentum

Hence momentum is conserved.