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 u_{1} and v_{1},
respectively collides with another body of mass m2, initial and final velocities
u_{2} and v_{2}, respectively, then according to the law of
conservation of momentum, we have,
m_{ 1}u_{1} + m_{2}u_{2} = m_{1}v_{1}
+ m_{2}v_{2}
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 m_{1}, initial and final velocities,
u_{1} and v_{1} respectively, with another body of mass m_{2},
initial and final velocities u_{2} and v_{2} respectively, both
the momentum and kinetic energy are conserved and can be expressed as follows:
m_{1}u_{1}
+ m_{2}u_{2} = m_{1}v_{1} + m_{2}v_{2 }
momentum conserved
^{1}/_{2}
m_{ 1}u_{1}^{2} + ^{1}/_{2}m_{2}u_{2}^{2}
= ^{1}/_{2}m_{1}v_{1}^{2} + ^{1}/_{2}m_{2}v_{2}^{2 }
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_{
1}u_{1} + m_{2}u_{2} = m_{1}v + m_{2}v
m_{ 1}u_{1}
+ m_{2}u_{2} = (m_{1} + m_{2})v
 momentum conserved
^{1}/_{2}
m_{ 1}u_{1}^{2} + ^{1}/_{2}m_{2}u_{2}^{2}
= ^{1}/_{2}m_{1}v^{2} + ^{1}/_{2}m_{2}v^{2
}
^{ 1}/_{2}
m_{ 1}u_{1}^{2} + ^{1}/_{2}m_{2}u_{2}^{2}
= ^{1}/_{2}(m_{1} + m_{2})v^{2 }
 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}/_{2}mu^{2}
respectively will be zero as well.
Therefore,
if a body of mass m_{1}, with initial velocity u_{1}, collides
with another body of mass m_{2} 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_{ 1}u_{1} + 0 = m_{1}v
+ m_{2}v
m_{ 1}u_{1} = (m_{1}
+ m_{2})v
Conservation of kinetic energy 
^{
1}/_{2} m_{ 1}u_{1}^{2} + 0 = ^{1}/_{2}m_{1}v^{2
}+ ^{1}/_{2}m_{2}v^{2}
^{
1}/_{2} m_{ 1}u_{1}^{2} = ^{1}/_{2}(m_{1}^{
}+ m_{2})v^{2}
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.
See calculations based on the law of conservation of momentum.
