Physics I
Class 06
Conservation of Momentum
Rev. 14-Jan-06 GB
06-1
What is a System of Objects?
The universe is too large to include all of it
in an experiment. We can only concentrate
our attention on a small part. If we do
things right, we can select a small group of
interacting objects in such a way that the
phenomenon we want to study is not
significantly influenced by anything else.
How to “do things right” is the tricky part.
A “system of objects” is a subset of the universe
that we have selected to study a phenomenon.
06-2
Internal and External Forces
Our system here consists of Objects A and B.
Forces between A and B are internal forces.
Forces on A or B from sources outside the system are external forces.
If we change the definition of the system, could that affect which forces
are internal and which are external?
F on A from C
F on B from C
External Forces
F on A from B F on B from A
Object A
Internal Forces
Object B
06-3
The Momentum of a System
The momentum of a system is the sum of all the individual parts:
 N 
P   pi
i 1
Newton’s Second Law for each object:


 d pi
Fnet ,i  m i a i 
dt
Newton’s Second Law for the system:
 N 

dP
d pi N 

  Fnet ,i   F
d t i 1 d t i 1
all system
06-4
Cancellation of Internal Forces
Some forces in a system are internal, some are external.



 F   Fint   Fext
all system
The internal forces are all in Newton’s Third Law Pairs
within the system, so they sum exactly to zero in the system.



 F  0   Fext  Fext
all system
06-5
Conservation of Momentum
(in a Nutshell)
Only external forces can change the momentum of a system.


dP
  Fext
dt
If the external forces cancel and/or can be neglected, then momentum
is constant (zero time derivative), or as physicists say, conserved.

dP
0
dt
06-6
One-Dimensional Example
Two Carts on a Track
Two objects are initially at rest, P = 0.
The objects spring apart; the spring force is internal to the system.
After the spring pushes them apart, because P is conserved:
p1
p2
Pafter  Pbefore
P  p1  p 2  0  m1v1  m 2 v 2
m1v1   m 2 v 2
06-7
Conservation of Momentum in
Multiple Dimensions
 Each direction of motion is independent.
 Conservation of momentum occurs (or not) separately in each direction.
d Px
  Fext , x
dt
d Py
dt
  Fext , y
d Pz
  Fext ,z
dt
06-8
Collisions in Multiple Dimensions
Y
X
Before:
After:
Px ,before  m1v1, x ,before  m 2 v 2, x ,before
Px ,after  m1v1, x ,after  m 2 v 2, x ,after
Py ,before  m1v1, y ,before  m 2 v 2, y ,before
Py ,after  m1v1, y ,after  m 2 v 2, y ,after
06-9
Solving Multi-Dimensional
Two-Body Collision Problems
That sounds complicated. Even with only two bodies in two dimensions,
there are 8 components of velocity to consider.
However:
 Momentum is conserved (or not) in each direction separately.
 Conserved directions of momentum do not mix with other directions.
So all we really need to do is to solve two one-dimensional problems.
If conserved:
m1v1, x ,before  m 2 v 2, x ,before  m1v1, x ,after  m 2 v 2, x ,after
m1v1, y ,before  m 2 v 2 , y ,before  m1v1, y ,after  m 2 v 2 , y ,after
06-10
Center of Mass
Center of mass defined for a system:
N
M   mi
i 1
x cm
y cm
1 N
  mi x i
M i 1
1 N
  mi yi
M i 1
06-11
Center of Mass
Example
Y
6
x cm
m = 4 kg
center of mass
2
y cm
m = 2 kg
1

4  2  2  8   4
42
1

4  6  2  2   4 2
42
3
X
2
8
06-12
Velocity of the Center of Mass
System Momentum
Velocity of center of mass:

v cm

v cm


N
d x cm 1
d xi 1 N
1 N 


  mi
  mi vi   pi
dt
M i 1
d t M i 1
M i 1

N


1
P

P

M
v
  pi 
or
cm
M i 1
M
If the momentum of a system is conserved (constant),
so is the velocity of the center of mass.
06-13
Class #6
Take-Away Concepts (Pt. 1)
1.
2.
Systems; internal/external forces in systems.
Momentum defined for a system:
 N 
P   pi
i 1
3.
Newton’s
 Second Law for a system:

dP
  Fext
dt
4. Conservation
of momentum when


dP
  Fext  0
dt
Pafter  Pbefore
06-14
Class #6
Take-Away Concepts (Pt. 2)
5.
6.
7.
Momentum is conserved (or not) separately for each direction
if Fext for that direction is negligible (or not).
Conserved components of momentum do not mix with each
other.
Center of mass defined (x equation for example):
x cm
8.
1 N
  mi x i
M i 1
Velocity of the center of mass and system momentum:


P  M v cm
06-15
Class #6
Problems of the Day
___ 1. In the figure below, penguin A is at the right edge of a uniform sled which
lies on the frictionless ice of a frozen lake, initially at rest. Penguin A sees
penguin B on the shore of the lake, and waddles to the left edge of the sled to
get closer, as indicated by the arrow.
While penguin A is waddling from the right edge to the left edge of the sled,
the center of mass of the system made up of penguin A and the sled
A. is moving toward the shore.
B. is moving away from the shore.
C. remains the same distance from shore.
D. could go either way, depending on the masses of the penguin and the sled.
E. who cares, they are only penguins.
06-16
Class #6
Problems of the Day
2. A hockey puck sliding on frictionless ice (the best kind) collides
with a second puck that is initially at rest. Both pucks have the same
mass. Define the +X direction as the direction of the initial velocity
of the first puck and +Y as 90º counter-clockwise from +X. After
the collision, the first puck has a velocity of 2.0 m/s in a direction
30º counter-clockwise from +X. What is the Y component of the
velocity of the second puck? (include + or – sign)
06-17
Activity #6 - Conservation of
Momentum in Two Dimensions
Objectives of the Activity:
1.
2.
3.
Use LoggerPro-3 and a video clip to study conservation of
momentum for a two-object system in two dimensions.
Practice calculating and understanding the center of mass.
Practice solving two-dimensional conservation of momentum
problems.
06-18