Momentum
Inertia in Motion
Definition of Momentum• Momentum is the product of mass and
velocity.
p mv
Momentum mskg Mass kg Velocity m
s
Momentum Depends on Velocity• An object must be moving in order to have
momentum.• Consider the truck at rest.• Consider the skate boarder. He has momentum.
0Truckp
0Skateboarderp
Momentum Depends on Mass• Larger objects traveling at the same velocity as
smaller objects will have greater momentum.
ManTruck pp
The Nature of Momentum• Momentum is a vector quantity.
– It has magnitude (how big).– It has direction (right(+) or left(-)).
• Direction is denoted by sign (+ or -).
0Truckp 0Busp
Collisions• The collisions that we will study will involve two objects.• Each object possesses its own momentum.• Collisions are broken down into two categories:
– Elastic (Objects Separate)– Inelastic (Objects Stick Together)
Momentum – Elastic Collisions• Elastic Collision – a collision in which the colliding bodies
do not stick together.• The equation used for elastic collisions is as follows.
FFII vmvmvmvm 22112211
Momentum – Inelastic Collisions• Inelastic Collision – a collision in which the colliding bodies
stick together.• The equation used for inelastic collisions is as follows.
1 1 2 2 Fm v m v MV
Elastic Collisions• The total momentum of the system remains
constant.
• Elastic collision, objects move off with separate velocities.
• This means that they can bounce.• Or they can be projected from rest (like a cannon).
Head on Elastic collisions
( ')( ') ( ' )( ' )a a a a a b b b b bm v v v v m v v v v
' 'a a b b a a b bm v m v m v m v
' 'a a a a b b b bm v m v m v m v
( ') ( ' )a a a b b bm v v m v v
In Elastic collisions
Head on Elastic collisions
( ')( ') ( ' )( ' )
( ') ( ' )a a a a a b b b b b
a a a b b b
m v v v v m v v v v
m v v m v v
( ') ( ' )a a b bv v v v
In Elastic collisions, KE is conserved
2 2 2 21 1 1 12 2 2 2' 'a a a a b b b bm v m v m v m v
2 2 2 21 1 1 12 2 2 2' 'a a b b a a b bm v m v m v m v
2 2 2 2( ' ) ( ' )a a a b b bm v v m v v 2 2( ) ( )( )a b a b a b
Impulse (mv)• When objects collide, their velocities
change. This means that their individual momentums will also change.
• Impulse simply defines the change in momentum for an individual object (denoted by mv).
Impulse (mv)• There are two ways to calculate impulse:
– You can subtract initial momentum from final momentum.
– Or you can take the product of force and time.
• The units for impulse are in Ns, which is equal to kgm/s
After BeforeJ mv p p
J mv Ft
End Chapter 7
Sample Problem 1 (Impulse)• A golf ball, initially at rest, is struck by a
golf club.
– The golf ball (m = 0.1kg) rolls away with a velocity of 35m/s.What is the initial momentum of the ball?What is the final momentum of the ball?What is the impulse on the ball?
0.1 0 0m ms sBefore Ip mv kg kg 0.1 35 3.5m ms sAfter Fp mv kg kg
After Beforep p p
3.5 0 3.5m m ms s sp kg kg kg
03.53.5
Swing
Practice Problem (Impulse)• A soccer ball (m = 0.4kg) approaches a player with a
velocity of 13m/s (left). The player kicks the ball giving it a new velocity of 20m/s (right).What is the initial momentum of the soccer ball?What is the final momentum of the soccer ball?What is the impulse felt by the soccer ball?
0.4 13 5.2m ms sp mv kg kg 0.4 20 8.0m m
s sp mv kg kg
8.0 5.2
13.2
F I
m ms s
ms
p p p
p kg kg
p kg
Kick
5.2 msIp kg
8.0 msFp kg
Ballistic pendulum• The ballistic pendulum has two parts.
Conservation of MomentumConservation of Energy
Ballistic pendulum
0 ( ) 'mv m M v
212 ( ) ' ( )m M v m M gh
( ) 'm M vv
m
Conservation of Momentum
Conservation of Energy
KE GPE21
2 'mv mgh
' 2v gh
( )2
m Mv gh
m
Sample Problem (Elastic)
FFII vmvmvmvm 22112211
mcannon = 300kg
mball = 10kg
Fire!
Momentum
• Identify the number and types of collisions in the animation below.
Practice Problem 3 (Elastic)• Two air track cars collide and bounce.• The red car is initially traveling at a velocity of 0.50m/s to the right
while the yellow car is traveling at a velocity of 1.5m/s to the left.• After the collision the red car is moving 2.0m/s to the left with the
yellow car following, but not touching.
Air Track0.3kg 0.8kg
mred = 0.3kg
myellow = 0.8kgFFII vmvmvmvm 22112211
What is the initial momentum of the red car?What is the initial momentum of the yellow car?What is the final momentum of the red car?What is the final momentum of the yellow car?
What is the final velocity of the yellow car?
0.3 0.5 0.15m ms sp mv kg kg 0.3 2.0 0.6m m
s sp mv kg kg 0.8 1.5 1.2m ms sp mv kg kg
0.15 0.61.2
0.15 1.2 0.6 ?
? 0.45 mskg
2
0.45
0.8
ms
F
kgv
kg
2 0.5625m
sFv
0.45
pp mv v
m
Collide
Practice Problem 4 (Elastic)• A gun (m = 1.5kg), initially at rest, is fired sending its
bullet (m = 0.1kg) at a velocity of . 180 ms
Before
After
What is the initial momentum of the gun and bullet?What is the total momentum of the system before firing?What is the total momentum of the system after firing?What is the final momentum of the bullet?What is the final momentum of the gun?What is the final velocity of the gun?
FFII vmvmvmvm 22112211 0 0 18 18
0.1 180 18m ms sp mv kg kg
1812
1.5
ms m
s
p kgp mv v
m kg
1.5 0 0
0.1 0 0
msGun
msBullet
p mv kg
p mv kg
00
0 0 18
18
x
x
Homework
• WS1 – #’s 1,2, 4-17
Inelastic Collisions• The Law of Conservation of Momentum: that the total
momentum of the system remains constant, no matter what the collision.
• In an inelastic collision, objects stick together.
• This means that they have the same final velocity (vF).
S
N
v2vF vF
Collide
Inelastic Collision Equation• The inelastic collision equation takes
into account the momentum of each object before and after the collision.
• Because the objects are combined after the collision, and they have the same final velocity (vF).
Before After
Inelastic Collision Equation
TotAfterTotBefore pp
FII vmmvmvm 212211
Initial Momentum Object 1
Initial Momentum Object 2
Final Combined Momentum
Objects 1 & 2
Sample Problem (Inelastic)The big rig backs up with an initial velocity of 2.0m/s and collides with the trailer, which is initially at rest.
FII vmmvmvm 212211 What is the initial momentum of the truck?What is the initial momentum of the trailer?What is the combined mass of the truck and trailer?What is the final velocity of the truck and trailer combo?
mTruck = 2000.0kg
mTrailer = 3000.0kg
003000 kgmvp sm
sm kgkgmvp 40000.22000 kgkgkgmm 50003000200021 Fs
m vkgkg 500004000
kg
vkg
kg
kg Fsm
5000
5000
5000
4000 s
mFv 8.0
04000 5000
BeforeAfter
0.8
ICE
Practice Problem 5 (Inelastic)• Happy Penguin (m = 7.0kg) slides across the ice on a magnet
with a velocity of 3.0m/s. • Cool Penguin (m = 8.0kg) comes up from behind with a velocity
of 5.0m/s. He, also having a magnet, will collide and stick.
FII vmmvmvm 212211
S
N
C
N
S
H
What is the initial momentum of Cool Penguin?What is the final momentum of Happy Penguin?What is the combined mass of the two penguins?What is the final velocity of the penguin pair?
sm
sm kgkgmvp 0.210.30.7 s
ms
m kgkgmvp 0.400.50.8 kgkgkgmm 0.150.80.721
0.15
0.15
0.15
0.61 Fv
sm
Fv 07.4
Fv0.150.61
40.021.0 15.0 4.07
Collide
Momentum While climbing a cliff, a super model (m =
51.0 kg) slips and falls. She falls for 2s before she is rescued by
Super Doctor Physics (m = 63.0 kg, v = 27.85 m/s).
What was their velocity immediately after the collision?
Homework
• WS 2– # 1,2
• WS 3– #1
Sample Problem 2 (Impulse)• A baseball is struck by a bat with a force of
3000N for a duration of 0.05s. What is the change in momentum of the baseball?
p Ft
150p Ns 3000 0.05p N s
150 msp kg
or
NF 3000st 05.0
Hit
Practice Problem 7 (Impulse)• A tennis ball is struck with a force of 400N
for a time of 0.15s.What is the impulse felt by the tennis ball?
p Ft 400 0.15 60p N s Ns
Hit
Summary• All problems surrounding momentum are
based on the following principles:– Momentum depends on both mass and velocity.– Momentum is conserved, regardless of collision.– Collision type determines the equation used.– Impulse is simply change in momentum.
Trains are stupid cuz they can’t do
Physics!