Transcript Mechanics 105 chapter 8

Momentum and collisions
(chapter eight)
 Linear
momentum and its
conservation
 Impulse and momentum
 Collisions
 Two-dimensional collisions
 Center of mass
 Systems of particles.
Linear Momentum
Linear momentum is defined as the mass times the
velocity of an object:


p  mv
This is a vector equation - in terms of the Cartesian
coordinates:
p x  mvx
p y  mv y
p z  mvz
Linear momentum
The time rate of change of momentum of a
particle equals the net force acting on it


 dp

dv
 F  dt  m dt  ma
Momentum conservation
Consider an isolated system of two-particles,
interacting with each other.
The force on object 1 exerted
 by object 2 is

dp1
F21 
dt
The reaction force to this isthe force on 2 by 1

dp2
F12 
dt
Since these make up an action-reaction pair




dp1
dp 2
d  
F21   F12 

  p1  p2   0
dt
dt
dt
ConcepTest
Momentum conservation
 The
result is a basic statement of the law of
conservation of the total momentum for an
isolated system.
 We can use this result to solve many types
of collision and interaction problems in
which we can choose the system to include
two (or more) interacting objects.
 Momentum is a vector, so its conservation
holds separately for each component
Examples
Two carts (masses m1 and m2) are held together at
rest on a frictionless cart with a spring between
them. When released, what are their velocities?
m1
Before
m2



pi  m1v1i  m2v2i  0

v1
After

v2



p f  m1v1 f  m2 v2 f  0

m1 
v2 f  
v1 f
m2
Conservation of momentum
 Examples
 Demo
 ConcepTest
Impulse and momentum
The change in momentum of an object during a collision
is caused by a net force. Integrating the differential for the
momentum:
t
t


p   dp 
f
ti

  Fdt  I
f
ti
This quantity (the net force integrated over time) is called
impulse, and results in another statement of Newton:s
2nd law: The total impulse of the net force equals the
change in momentum of the particle.
Impulse approximation
tf


The time average of the force is defined as F  1 Fdt
t ti
So the impulse momentum theorem can be written


I  p   F t
In the impulse approximation, we assume one of the
forces acting on the object is much larger than the
others, but acts only for a small interval.
Collisions
Use the impulse approximation - forces of collisions
much larger, shorter duration than other forces on
objects.
Two extremes of collisions:
Elastic - kinetic energy conserved (e.g. billiard balls)
Perfectly inelastic - object stick together (maximum
KE loss)
Momentum is conserved in both
Collisions, cont.
 One
dimensional elastic
Momentum
m1v1i  m2v2i  m1v1 f  m2v2 f
conservation
1
1
1
1
2
2
2
2
m1v1i  m2v2i  m1v1 f  m2v2 f
Energy
2
2
2
conservation 2
After lots of algebra
 m1  m2 
 2m2 
v1i  
v2i
v1 f  
 m1  m2 
 m1  m2 
v2 f
 2m1 
 m2  m1 
v1i  
v2i
 
 m1  m2 
 m1  m2 
2-d Collisions
Momentum conservation – vector equation – holds for
each component
For elastic collisions – conservation of kinetic energy
(magnitude of the velocity)
Example
Center of mass
= the average position of a system’s mass
in the case of discrete point objects 
rCM 

 mi ri
i
m
i
for an extended object

rCM 

r
 dm
i
1 

r dm

 dm M





The white part if your fingernail is called the lunia.
No U.S. President has had brown eyes.
In the year 2160, there will be two lunar eclipses and
five solar eclipses.
Benjamin Franklin invented the rocking chair.
3 out of 4 people make up 75% of the population.
Motion of a system of particles
= the velocity of the center of mass of a
system of particles is just



drCM
dri
1
1
vCM 

mi


dt
M i
dt M



 ptot   pi  MvCM

 mi vi
i
i
in other words, the total momentum is just the
total mass times the velocity of the center of
mass. The acceleration of the COM is


dvCM
1
aCM 

dt
M


  Fi  MaCM
i

dvi
1
m

i i dt M

m
a
 ii
i
Motion of a system of particles
The forces Fi has terms that are internal and
external, but the internal forces sum to zero
by Newton’s 3rd law




dptot
F

F

M
a

 ext i i
CM
dt
So the system moves like a particle of mass
M located at the COM.
If the net external force is zero, the total
momentum of the system is conserved
The hardness of butter is directly proportional
to the softness of the bread.
 To steal ideas from one person is plagiarism;
to steal from many is research.
 For every action there is an equal and
opposite criticism.
 If the speed of light is 186,000 miles/sec.,
what's the speed of darkness?
 The sooner you fall behind the more time
you'll have to catch up.

 http://www.mistupid.com/psych/proftest.ht
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