Conservation of Linear Momentum
System:
Closed System:
Isolated System:
In a closed, isolated system, the net linear momentum is
Momentum is a
For example:
and is not necessarily conversed in all directions.
Within a two object system:
You must use vector components and work with only one direction at a time, using appropriate algebraic
signs.
Examples:
Recoil:
Exploding bomb:
Thrust:
Collisions
What is a collision?
A. “A collision is an isolated event is which a relatively strong force acts on each of two or more
colliding bodies for a relatively short time.”
B. It is important to carefully make a distinction between before, during, and after a collision.
C. Conservation of linear momentum only applies when only internal forces act.
D. A “crash” or “physical contact” is NOT required. For example: a space probe approaching a
planet encounters a sling-shot encounter. The “collision” force is gravity!
E. Virtually all of our knowledge of the sub-atomic world comes from creating and observing the
effects of collisions.
F. Conservation of Energy only applies to special cases of collisions.
G. VERY IMPORTANT: the motion of the center of mass is unaffected by a collision!
Inelastic Collisions
Virtually all collisions in the macroscopic world are inelastic. Kinetic energy is lost in the form of heat
and sound, yet linear momentum will be conserved if the system is closed and isolated. When
objects stick together after the collision, the collision is said to be “completely” or “totally” or
“perfectly” inelastic.
Inelastic collisions
Collisions in which kinetic energy is not conserved but momentum is conserved.
Assume ALL collisions are inelastic unless told otherwise.
Examples:
One- dimensional
A bullet passes through a stationary target
Completely inelastic
A bullet is imbedded in a stationary target
A “ballistic pendulum”: Conservation of linear momentum applies to the collision only. Conservation
of mechanical energy applies to the swing of the pendulum only.
A “reverse” ballistic pendulum
Two dimensional collisions
Intersections- objects approach each other at 90 degrees.
One object approaches a stationary object, but there is an “impact parameter”
If the two objects have the same mass, the angle between them after the impact will always be 90
degrees.
Elastic Collisions
In Elastic Collisions both
are conserved.
and
Elastic collisions very rarely occur except on the atomic level. However, collisions of billiard balls are
very close to elastic. Conservation of momentum and of kinetic energy will yield ____
valid
equations (which can be VERY tedious to deal with).
When a collision is elastic, using a
greatly simplify collision problems.
Recall- when the net external force on a system is zero,
reference frame will
therefore the velocity of the
is constant. However, from the frame of reference of the cm, that velocity is __________
and so, in the cm frame, the total momentum which equals
is also equal to
So… for a collision between two objects, in the cm frame, their individual momenta before the collision
must be
to their individual momenta after the collision.
Example: A 4 kg block moving right at 6 m/s collides elastically with a 2 kg block moving right at 3
m/s. What are their final velocities? (wow, TWO unknowns! But, don’t worry, the cm frame makes this
easy!)
1. Calculate the velocity of the center of mass.
2. Transform each initial velocity to the cm frame (by subtracting the velocity of the cm).
3. Reverse the velocity of each object to solve for each one’s final velocity.
4. To find the final velocities in the original frame, add the velocity of the center of mass back into
each final velocity.