Linear momentum and the impulse-momentum
theorem
Definition of linear momentum of a particle
Consider a system of N particles, each with a position vector, ri .
The ith particle has mass mi and velocity, vi . We define the linear
momentum of the particle by
pi  mivi
.
The linear momentum of the system of particles is then written
P  i pi .
Next, we contemplate how to change the value of p for a particle.
Obviously, the mass can not change and therefore to change p we
must do it by changing v, i.e we must accelerate the particle.
Newton’s second law gives
Fnet = m dv/dt = d [mv]/dt = dp/dt.
This becomes for a system of particles,
Fextnet = dP/dt,
Where, Fextnet is the sum of only the external forces acting on the
particles of the system.
Rearranging gives the impulse-momentum theorem,
Fnet dt = p,
where the impulse of a force is defined,
I   F dt.
this is the area under F vs t graph
For example,
Fnet
t = Fnet
t
= p where the average force is
used.
A consequence of the impulse-momentum theorem is the law of
conservation of momentum:
If the net external force acting on a system is zero, then the
momentum of the system is constant.
An important application is to collisions. In the simplest case we
consider the collision of two particles in one-dimension, of which
there are two extreme regimes.
 completely inelastic
Only momentum of the two-body system is conserved.
 Elastic
Both momentum and kinetic energy of the two-body system is
conserved.
EXAMPLES [in class]