Physics I
Class 10
Conservation of Momentum
in Two and Three Dimensions
10-1
Class #9
Take-Away Concepts (Review)
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
10-2
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
10-3
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
10-4
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
10-5
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
10-6
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
10-7
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.
10-8
Class #10
Take-Away Concepts
1.
2.
3.
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
4.
1 N
  mi x i
M i 1
Velocity of the center of mass and system momentum:


P  M v cm
10-9
Activity #10 - Conservation of
Momentum in Two Dimensions
Objectives of the Activity:
1.
2.
3.
Use VideoPoint 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.
10-10