The linear impulsemomentum relation
Linear momentum of
a particle:
The symbol L denotes the
linear momentum and is defined as the mass times the velocity of a particle.
_{} 

Impulse of a force
from time t_{1 }to t_{2}: The integral of the force
over the time interval of concern is its impulse. The impulse of a force is a
vector given by the integral

_{}
Newton’s 2^{nd}
law:
Newton’s second law states that the resultant of all forces applied on a
particle is equal to the rate of change of linear momentum of the particle.
_{}
This
reduces to the more familiar statement of this law if one notes that in
Newtonian mechanics it is assumed that mass is constant, and, therefore, _{}.
The linear
impulsemomentum relation: Integration of Newton’s 2^{nd} law over the time interval
from t_{1} to t_{2} results in
_{}
_{}
Therefor,
the linear momentum of a particle is changed by the impulse of the resultant
force on the particle. There will be conservation
of linear momentum only if the impulse of the resultant force is zero.
For a system of
particles:
Consider the system of particles shown below. Each particle in the system has a
mass m_{i} and at time t_{1} has a velocity v_{i}, and from time t_{1} to time t_{2} is acted upon by the resultant external force F_{i} and the internal forces of interaction between
the particles of f_{ij}. As a result of the impulse of the internal
and external forces on each particle, at time t_{2} the particle with mass m_{i} has a velocity v^{*}_{i}.

Writing
the impulse momentum equation for each particle and adding them up, considering
that Newton’s 3^{rd} law requires that _{}, we get for n
particles
_{}
Therefore,
the total linear momentum of the system is changed by the impulse of the
external forces. Recalling the relation _{}, we can rewrite this equation as
_{}
ã Mehrdad Negahban and the
University of Nebraska, 19962002.
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Department
of Engineering Mechanics, University of Nebraska, Lincoln, NE 685880526