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Apply the impulse-momentum theorem to m1 and m2 separately,
t
= p1 = m1v1f- m1v1i
t
= p2 = m2v2f- m2v2i
where = the average force ofm2 on m1 , and = the average force
ofm1 on m2 . By
Newton's third lawF1(t) = -F2(t) which gives = - and so,
( + ) t= m1v1f- m1v1i + m2v2f- m2v2i = 0
p1f+p2f =p1i +p2i. (4)
This is the statement of the conservation of momentum.
Note:
y The system must be isolated: the affect of all external forces
actingon m1 and m2 must be negligable.
y The conservation of momentum holds for a collision involving
any number of
objects:
= .
(5)
y
Momentum is a vector, and each component is conserved
separately. The equationfor conservation of momentum really
contains three equations, one for eachdimension.
y From Newton's third law of motion we know that whenever a
force is applied
on a body there will be an equal and opposite reaction. Action
and reaction
forces result in change in velocities of both the bodies which
in turn change
the momenta of these bodies.
y In an elastic collision the initial momentum of the bodie
found to be equal to the final momentum of the bodies
y Thus Newton's second and third laws of motion lead u
law of mechanics, the law of conservation of momentu
y Law of conservation of momentum states that - if a gro
exerting force on each other, i.e., interacting with each
momentum remains conserved before and after the in
is no external force acting on them.
y The following example will help us to understand clear
conservation of momentum.
y Two bodies A and B of masses m1 and m2 are moving
with initial velocities u1and u2 . They make a direct coll
that after collision they continue moving in the same d
collision last for a very short interval of time 't' seconds
y During collision, A exerts a force on B. At the same tim
A. Due to these action and reaction forces the velocitie
changed. After collision, let v1 and v2 be the velocities
respectively.
y The force exerted on A = m1 a1
y [According to Newton's II law of motion]
y
y
y
8/8/2019 Momentum P6
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y
y
y
y Therefore, force exerted on B = m2a2
y
y According to Newton's third law of motion, these two
forces
are equal and opposite.
y i.e., F1 = -F2
y
y
y
i.e., total momentum before collision is equal to the total
momentum after
collision, which is nothing but law of conservation of
momentum.
y Conservation of momentum is a fundamental law of physics which
states that
the momentum of a system is constant if there are no external
forcesacting
on the system. It is embodied in Newton's first law (the law of
inertia).
y Suppose we have two interacting particles 1 and 2, possibly of
different
masses. The forces between them are equal and opposite.
According
toNewton's second law, force is the time rate of change of the
momentum, so
we conclude that the rate of change of momentum of particle 1 is
equal to
minus the rate of change of momentum of a particle 2,
(1)
y Now, if the rate of change is always equal and opposite, it
follows that the
total change in the momentum of particle 1 is equal and opposite
of the total
change in the momentum of particle 2. That means tha
momenta the result is zero,
y But the statement that the rate of change of this sum is
stating that the quantity is a constant. This su
momentum of a system, and in general it is the sum of
momenta of each particle in the system.
y For electromagnetic radiation,
so in cgs,
y
where T is the Maxwell stress tensor, is the force d
the Poynting vector, c is the speed of light, and
density.Examples of Momentum Conservation
Imagine a sailboat that can't get where it's going because
the
the captain sets up a huge fan on the boat to blow wind into
thsailboat move? A large enough fan that is not on the boat cou
the fan were on the boat the boat would not go anywhere. Th
constitute a closed system, so the total momentum can not ch
The recoil from a gun and rocket propulsion are also example
momentum. The initial momentum is zero. The bullet or rockein
one direction. To keep the total momentum zero, the rocke
same momentum in the opposite direction.
