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Momentum

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The Law Of Conservation Of Momentum

From Newton's Second Law of Motion, an important law of conservation, known as the Law of Conservation of Momentum can be formulated. The law states:

In a system of colliding objects the total momentum is conserved, provided there is no net external force acting on the system.

The law of conservation of momentum can also be expressed in this way:

The total momentum of an isolated or closed system of colliding bodies remains constant.

Therefore, if two or more bodies collide in a closed system, the total momentum before collision is equal to the total momentum after collision.

Note that Closed or Isolated system means a system that is not acted upon by any external force.

If a body of mass m1, initial and final velocities u1 and v1, respectively collides with another body of mass m2, initial and final velocities u2 and v2, respectively, then according to the law of conservation of momentum, we have,

m 1u1 + m2u2 = m1v1 + m2v2

The law of conservation of momentum is made possible when body A collides with B, because the Action of A on B is equal to the Reaction of B on A, and both forces act precisely for the same time.

The Law Of Conservation Of Momentum For Elastic And Inelastic Collisions

Here, let's see how the law of conservation of momentum is expressed for the two types of collisions - Elastic and Inelastic Collisions.

Elastic Collision

Elastic collisions are collisions in which both momentum and kinetic energy are conserved.

For the elastic collision of a body of mass m1, initial and final velocities, u1 and v1 respectively, with another body of mass m2, initial and final velocities u2 and v2 respectively, both the momentum and kinetic energy are conserved and can be expressed as follows:

m1u1 + m2u2 = m1v1 + m2v2          - momentum conserved

1/2 m 1u12 + 1/2m2u22 = 1/2m1v12 + 1/2m2v22     - kinetic energy conserved

A ball that hits and bounces off the ground to its original height is an example of perfectly elastic collision.

Inelastic Collision

An inelastic collision is a collision where momentum is conserved, but kinetic energy is not.

Here, kinetic energy actually decreases as it is turned into certain other energies, such as sound, heat, or elastic potential energy, during the collision.

When two bodies collide and stick together, moving with the same velocity afterward, such collision is said to be completely inelastic. Both the conservation of momentum and kinetic energy can be expressed as follows:

m 1u1 + m2u2 = m1v + m2v

m 1u1 + m2u2 = (m1 + m2)v     - momentum conserved

1/2 m 1u12 + 1/2m2u22 = 1/2m1v2 + 1/2m2v2

1/2 m 1u12 + 1/2m2u22 = 1/2(m1 + m2)v2      - kinetic energy conserved

Now what about the situation whereby one body was at rest before it was collided upon by another body, how would the conservation of their momentum and kinetic energies be expressed?

Here is how.

A body that is at rest has its initial velocity, u, to be zero. That means both its initial momentum and kinetic energy, mu and 1/2mu2 respectively will be zero as well.

Therefore, if a body of mass m1, with initial velocity u1, collides with another body of mass m2 at rest and both bodies join and move together with velocity v, then their conservation of momentum and kinetic energies can be stated as follows:

Conservation of momentum -  m 1u1 + 0 = m1v + m2v

m 1u1 = (m1 + m2)v

Conservation of kinetic energy -

1/2 m 1u12 + 0  = 1/2m1v2 + 1/2m2v2

1/2 m 1u12   = 1/2(m1 + m2)v2

Note: In working out calculations based on the law of conservation of momentum, it is important that the directions of the velocities must be considered along the same line with positive or negative signs.