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Case 1Increasing Momentum
Follow through
Examples:Long Cannons
Driving a golf ballCan you think of others?
t t p
I
F
I
p
Case 2Decreasing Momentum over a Long Time
Examples:
Bungee Jumping
Can you think of others?
Ip
tF
tF
Warning – May be dangerous
Case 3Decreasing Momentum over a Short Time
Examples:
Boxing (leaning into punch)
Head-on collisions
Can you think of others?
tFIp
5. CONSERVATION OF MOMENTUM
Example:Rifle and bullet
Demo - Rocket balloon Demo - Clackers
Video - Cannon recoilVideo - Rocket scooter
Consider two objects, 1 and 2, and assume that no external forces are acting on the system composed of these two particles.
ifvmvmtF
11111
Impulse applied to object 1
ifvmvmtF
22222
ififvmvmvmvm
222211110
Impulse applied to object 2
Total impulseappliedto system
ffiivmvmvmvm
22112211
or
Apply Newton’s Third Law21
FF
tFtFor 21
Internal forces cannot cause a change in momentum of the system.
For conservation of momentum, the external forces must be zero.
6. COLLISIONS
Collisions involve forces internal to
colliding bodies.
Elastic collisions - conserve energy and
momentum
Inelastic collisions - conserve momentum
Totally inelastic collisions - conserve
momentum and objects stick together
Elastic and Inelastic Collisions
In an elastic collision the total kinetic energy is conserved Momentum is conserved in any
collision Example:
What are signs of final velocities?
22
212
1223
21
2112
2
and 22Let
ff
ff
ii
mvmvmvK
mmm
mmm
vvvP
vvv
Elastic and Inelastic Collisions
Example (cont.): Consider reference frame where CM is at rest
vvvvvvvv
vvvv
vv
vvvvP
vvvvvvvv
vvvv
31Lab
CMCM2235Lab
CMCM11
32
CM234
CM1
CM*CM*
CM2CM1CM2CM1
32Lab
CM2CM234Lab
CM1CM1
31
31Lab
CM
,
,
220
and ,
)2(
ffff
ff
if
iiii
iiii
m
mm
--
mm
Elastic and Inelastic Collisions
In an inelastic collision the total kinetic energy is not conserved Momentum is conserved
in any collision Example: case where
particles stick together
vv
vvvvP
vvv
31
21
2112
32
and 22Let
f
fff
ii
mmmm
mmm
Elastic and Inelastic Collisions
Example: Ballistic Pendulum
ghv
ghmmvmm
vv
vmmvm
mmm
A
B
Ammm
B
BA
2
)()(
)(
1
21
21
1
1
212
2121
1
2111
Collision between two objects of the same mass. One mass is at rest.
Collision between two objects. One not at rest initially has twice the mass.
Collision between two objects. One at rest initially has twice the mass.
Simple Examples of Head-On Collisions
(Energy and Momentum are Both Conserved)
Head-On Totally Inelastic Collision Example
Let the mass of the truck be 20 times the mass of the car.
Using conservation of momentum, we get
mphvtruck 60 mphvcar 60
vmmphmmphm )21()60()60(20
vmph 21)60(19
)60(21
19mphv
mphv 3.54
initial momentum of system = final momentum of system
Remember that the car and the truck exert equal but oppositely directed forces upon each other.
What about the drivers? The truck driver undergoes the same
acceleration as the truck, that is
t
mph
t
mph
7.5)603.54(
The car driver undergoes the same acceleration as the car, that is
t
mph
t
mphmph
3.114)60(3.54
The ratio of the magnitudes of these two accelerations is
207.5
3.114
Remember to use Newton’s Second Law to see the forces involved.
For the truck driver his mass times his acceleration gives
F
am
For the car driver his mass times his greater acceleration gives
ma
F
, big trucks that is. Your danger is of the order of twenty times
greater than that of the truck driver.
TRUCKS Don’t mess with T
Collisions in 2D
Use Conservation of Momentum in each direction Consider case where
one particle is at rest
In CM frame particles are back-to-back!
sinsin
coscos
2211
221111
ff
ffi
vmvm
vmvmvm
Rocket Propulsion
“Rockets can’t fly in vacuum. What do they have to push against?” Nonsense. Rockets
don’t push; they conserve momentum, and send parts (fuel) away from the body as fast as possible
Rocket Propulsion
How fast do rockets accelerate? Start at rest, with mass
M+m Some time t later, have
expelled m at speed ve, to conserve momentum, rest of rocket (M) must have velocity (in the other direction) of
v = ve m/M