Newton’s Third Law of Motion states that there is always reaction for every action in opposite direction and of equal magnitude.
Explanation: Whenever a force is applied over a body, that body also applies same force of equal magnitude and in opposite direction.
Law of Conservation of Momentum: The sum of momenta of two objects remains same even after collision.
In other words, the sum of momenta of two objects before collision and sum of momenta of two objects after collision are equal.
Mathematical Formulation of Conservation of Momentum:
Suppose that, two objects A and B are moving along a straight line in same direction and the velocity of A is greater than the velocity of B.
Let the initial velocity of A=u1
Let the initial velocity of B= u2
Let the mass of A= m1
Let the mass of B=m2
Let both the objects collide after some time and collision lasts for ' t' second.
Let the velocity of A after collision= v1
Let the velocity of B after collision= v2
We know that, Momentum = Mass × Velocity
Momentum of `A(F_A)` before collision `=m_1xxu_1`
Momentum of `B(F_B)` before collision `=m_2xxu_2`
Momentum of A after collision `=m_1xxv_1`
Momentum of B after collision `=m_2xxv_2`
Now, we know that Rate of change of momentum
=Mass x rate of change in velocity
=mass x Change in velocity/time
Therefore, rate of change of momentum of A during collision, `F_(AB)=m_1((v_1-u_1)/t)`
Similarly the rate of change of momentum of B during collision, `F_(BA)=m_2((v_2-u_2)/t)`
Since, according to the Newton's Third Law of Motion, action of the object A (force exerted by A) will be equal to reaction of the object B(force exerted by B). But the force exerted in the course of action and reaction is in opposite direction.
`=>m_1u_1+m_2u_2` `=m_1v_1+m_2v_2` ---(i)
Above equation says that total momentum of object A and B before collision is equal to the total momentum of object A and B after collision. This means there is no loss of momentum, i.e. momentum is conserved. This situation is considered assuming there is no external force acting upon the object.
This is the Law of Conservation of Momentum, which states that in a closed system the total momentum is constant.
In the condition of collision, the velocity of the object which is moving faster is decreased and the velocity of the object which is moving slower is increased after collision. The magnitude of loss of momentum of faster object is equal to the magnitude of gain of momentum by slower object after collision.
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