_ linear momentum and its conservation _ impulse and momentum _ collisions _ two-dimensional...
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Linear momentum and its conservation
Impulse and momentum Collisions Two-dimensional collisions Center of mass Systems of particles.
Momentum and collisions (chapter eight)
Linear Momentum
Linear momentum is defined as the mass times the velocity of an object:
This is a vector equation - in terms of the Cartesian coordinates:
vmp
zz
yy
xx
mvp
mvp
mvp
Linear momentumThe time rate of change of momentum of a
particle equals the net force acting on it
amdt
vdm
dt
Momentum conservationConsider an isolated system of two-particles,
interacting with each other.The force on object 1 exerted by object 2 is
The reaction force to this is the force on 2 by 1
Since these make up an action-reaction pair
ConcepTest
dt
pdF 1
21
dt
pdF 2
12
02121
1221 ppdt
d
dt
pd
dt
pdFF
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?
Before m1 m2
After
1v
2v
02211 iii vmvmp
ff
fff
vm
mv
vmvmp
12
12
2211 0
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:
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.
IdtFpdpf
i
f
i
t
t
t
t
Impulse approximation
The time average of the force is defined as
f
i
t
t
dtFt
F 1
So the impulse momentum theorem can be written
tFpI
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
222
211
222
211
22112211
2
1
2
1
2
1
2
1ffii
ffii
vmvmvmvm
vmvmvmvm
Momentum conservation
Energy conservation
After lots of algebra
iif
iif
vmm
mmv
mm
mv
vmm
mv
mm
mmv
221
121
21
12
221
21
21
211
2
2
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
for an extended object
ii
iii
CM m
rmr
dmr
Mdm
dmrrCM
1
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
CMi
itot
iii
i
ii
CMCM
vMpp
vmMdt
rdm
Mdt
rdv
11
= the velocity of the center of mass of a system of particles is just
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
CMi
i
iii
i
ii
CMCM
aMF
amMdt
vdm
Mdt
vda
11
Motion of a system of particlesThe forces Fi has terms that are internal and external, but the internal forces sum to zero by Newton’s 3rd law
dt
pdaMFF totCM
iiext
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.htm