Linear Momentum
The linear momentum p of an object
of mass m with a velocity of v is


p  mv
It is a vector and points in the same
direction as the velocity vector.
The momentum vector is an entirely
different vector than the velocity vector.
Care should be taken in comparing one to
the other.
It is safe to say that the momentum vector
is in the same direction as the velocity
vector as mentioned earlier.
One can also say that the momentum
vector is directly proportional to the
velocity vector, i.e., the momentum vector
doubles if the velocity vector doubles.
But momentum also depends on the mass.
So changing the mass of an object will also
change the momentum vector.
Therefore to change momentum one must
change the mass or velocity or both.
Regardless of what changes, the momentum
vector is always in the same direction as
the velocity vector.


p  mv
As long as there are no external
forces acting on a system of
particles, collisions between the
particles will exhibit conservation of
linear momentum.
This means that the vector sum of
the momenta before collision is equal
to the vector sum of the momenta of
the particles afterwards.


p  mv
This is known as the conservation of
linear momentum.
It is an extremely important concept
in physics.
One key area that makes use of this
conservation principle is collisions.
This is what you are going to explore
today.


p  mv
Simple Examples of Head-On Collisions
(Energy and Momentum are Both Conserved)
Collision between two objects of the same mass. One mass is at rest.
Collision between two objects. One at rest initially has twice the mass.
Collision between two objects. One not at rest initially has twice the mass.


p  mv
Simple Examples of Head-On Collisions
(Totally Inelastic Collision, only Momentum Conserved)
Collision between two objects of the same mass. One mass is at rest.
Collision between two objects. One at rest initially has twice the mass.
Collision between two objects. One not at rest initially has twice the mass.


p  mv
Example of Non-Head-On Collisions
(Energy and Momentum are Both Conserved)
Collision between two objects of the same mass. One mass is at rest.
If you do a vector addition of the two momenta after collision,
you find that it is equal to the total momentum before collision.


p  mv
Velocity Components in Projectile Motion
(In the absence of air resistance.)
Note that the horizontal component
of the velocity remains the same if air
resistance can be ignored.
The collision you will study will involve two
objects of equal mass colliding in a
non-head-on collision in a horizontal plane
and then undergoing projectile motion
after the collision.
Since the horizontal component of velocity
remains constant for a projectile in free
fall (shown in previous slide), the horizontal
part of the projectile motion can be used
to represent the horizontal component of
the momentum after collision.


p  mv
Here is an example of what you are
going to do in the exercise today.
You will roll a ball
down the curved
ramp.
z
This represents the velocity as the ball left the table
because the horizontal velocity of a projectile
remains constant in the absence of air resistance.


p  mv