Physics I
Class 08
Conservation of Momentum
in One Dimension
Rev. 11-Aug-03 GB
08-1
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.
08-2
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
08-3
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 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
08-4
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
08-5
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
08-6
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
08-7
Class #8
Take-Away Concepts
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
08-8
Class #8
Problems of the Day
_____1. You are investigating conservation of momentum in one
dimension by observing two carts – A and B – collide on a track, similar to
the carts and tracks we use in class. What should you pick for your
SYSTEM in order to have momentum conserved, at least approximately,
and also be able to do the experiment? (Pick all correct answers or leave
blank if none.)
A) Cart A only.
B) Cart B only.
C) Carts A and B, nothing else.
D) Carts A and B, the track, nothing else.
E) The universe.
08-9
Answer to Problem 1 for Class #8
The answers are C and D.
Each cart during a collision will experience a non-negligible force
from the other cart, and so in order to make sure those forces are
internal to the system, both carts should be included.
Including the track is optional, since it is not moving and we
assume that the friction of the carts on the track is negligible.
“The universe” is not a practical system to study conservation of
momentum unless you are a cosmologist.
08-10
Class #8
Problems of the Day
_____2. A rifle of mass M is initially at rest but free
to recoil. It fires a bullet of mass m and velocity +v
(relative to the ground) in the X dir. After firing, the
velocity of the rifle (relative to the ground) is:
A) –mv
B) –Mv/m
C) –mv/M
D) –v
E) +mv/M
08-11
Answer to Problem 2 for Class #8
The answer is C.
The system is the rifle and the bullet. The initial momentum of the
system is zero (at rest).
Momentum is conserved in this system because all external forces
are neglected. (The rifle is free to recoil.)
Let the velocity of the rifle be V. Since the total momentum after
firing is still zero, we have M V + m v = 0 or V = –mv/M.
08-12
Activity #8 Conservation of Momentum
Objectives of the Activity:
1.
2.
Think about how systems are defined and how that
affects the classification of internal/external forces.
Use VideoPoint to study conservation of momentum for
a two-object system in one dimension.
08-13