Momentum

Momentum

• From Newton’s laws: force must be present to change an object’s velocity (speed and/or direction)

Wish to consider effects of collisions and corresponding change in velocity

Method to describe is to use concept of linear momentum

scalar vector

Linear momentum = product of mass velocityLinear momentum = product of mass velocity

Golf ball initially at rest, so some of the KE of club transferred to provide motion of golf ball and its change in velocity

2212

21

221

ab

k

k

mvmvmvdv

mvdv

dE

mvE

Momentum

• Vector quantity, the direction of the momentum is the same as the velocity’s

• Applies to two-dimensional motion as well

yyxx mvpandmvp

vmp

Size of momentum: depends upon mass depends upon velocity

Impulse

• In order to change the momentum of an object (say, golf ball), a force must be applied

• The time rate of change of momentum of an object is equal to the net force acting on it

–

– Gives an alternative statement of Newton’s second law– (F Δt) is defined as the impulse– Impulse is a vector quantity, the direction is the same

as the direction of the force

tFporamt

vvm

t

pF net

ifnet

:)(

Graphical Interpretation of Impulse

• Usually force is not constant, but time-dependent

• If the force is not constant, use the average force applied

• The average force can be thought of as the constant force that would give the same impulse to the object in the time interval as the actual time-varying force gives in the interval

( )i

i it

impulse F t area under F t curve

If force is constant: impulse = F t

Example: Impulse Applied to Auto Collisions

• The most important factor is the collision time or the time it takes the person to come to a rest– This will reduce the chance of dying in a car crash

• Ways to increase the time– Seat belts– Air bags

The air bag increases the The air bag increases the time of the collisiontime of the collision and and absorbs some of the energyabsorbs some of the energy from the from the bodybody

ConcepTest

Suppose a ping-pong ball and a bowling ball are rolling toward you. Both have the same momentum, and you exert the same force to stop each. How do the time intervals to stop them compare?

1. It takes less time to stop the ping-pong ball.2. Both take the same time.3. It takes more time to stop the ping-pong ball.

Answer

Suppose a ping-pong ball and a bowling ball are rolling toward you. Both have the same momentum, and you exert the same force to stop each. How do the time intervals to stop them compare?

1. It takes less time to stop the ping-pong ball.2. Both take the same time.3. It takes more time to stop the ping-pong ball.

Note: Because force equals the time rate of change of momentum, the two balls loose momentum at the same rate. If both balls initially had the same momenta, it takes the same amount of time to stop them.

Problem: Teeing Off

A 50-g golf ball at rest is hit by “Big Bertha” club with 500-g mass. After the collision, golf leaves with velocity of 50 m/s.

a) Find impulse imparted to ballb) Assuming club in contact with

ball for 0.5 ms, find average force acting on golf ball

Problem: teeing off

Given:

mass: m=50 g = 0.050 kgvelocity: v=50 m/s

Find:

impulse=?Faverage=?

1. Use impulse-momentum relation:

2. Having found impulse, find the average force from the definition of impulse:

smkg

smkg

mvmvpimpulse if

50.2

050050.0

N

s

smkg

t

pFthustFp

3

3

1000.5

105.0

50.2,

Note: according to Newton’s 3rd law, that is also a reaction force to club hitting the ball:

iiff

ifif

R

VMvmVMvm

orVMVMvmvm

ortFtF

,

, of club

CONSERVATION OF MOMENTUM

Conservation of Momentum

• Definition: an isolated system is the one that has no external forces acting on it

– A collision may be the result of physical contact between two objects

– “Contact” may also arise from the electrostatic interactions of the electrons in the surface atoms of the bodies

Momentum in an isolated system in Momentum in an isolated system in which a which a collision occurs is conserved (regardless collision occurs is conserved (regardless of the nature of the forces between the of the nature of the forces between the objects)objects)

Conservation of Momentum

The principle of conservation of momentum states when no external forces act on a system consisting of two objects that collide with each other, the total momentum of the system before the collision is equal to the total momentum of the system after the collision

Conservation of Momentum

• Mathematically:

– Momentum is conserved for the system of objects– The system includes all the objects interacting with each

other– Assumes only internal forces are acting during the collision– Can be generalized to any number of objects

ffii vmvmvmvm 22112211

Problem: Teeing Off (cont.)

Let’s go back to our golf ball and club problem:

sm

kg

smkgvv

smkgvvm

smv

msmkgp

if

if

55.0

50.2

so,50.2:Club

50

gramm50,50.2:Ball

factor of 10 times smaller

ConcepTest

Suppose a person jumps on the surface of Earth. The Earth

1. will not move at all2. will recoil in the opposite direction with tiny velocity3. might recoil, but there is not enough information provided to see if that could happened

ConcepTest

Suppose a person jumps on the surface of Earth. The Earth

1. will not move at all2. will recoil in the opposite direction with tiny velocity3. might recoil, but there is not enough information provided to see if that could happened

Note: momentum is conserved. Let’s estimate Earth’s velocity after a jump by a 80-kg person. Suppose that initial speed of the jump is 4 m/s, then:

smkg

smkgV

smkgVMp

smkgp

Earth

EarthEarth

2324

103.5106

320

so,320:Earth

320:Person

tiny negligible velocity, in opposite direction

Types of Collisions

• Momentum is conserved in any collision

what about kinetic energy?

• Inelastic collisions– Kinetic energy is not conserved

• Some of the kinetic energy is converted into other types of energy such as heat, sound, work to permanently deform an object

– Perfectly inelastic collisions occur when the objects stick together

• Not all of the KE is necessarily lost

energylost fi KEKE

Perfectly Inelastic Collisions:

• When two objects stick together after the collision, they have undergone a perfectly inelastic collision

• Suppose, for example, v2i=0. Conservation of momentum becomes

fii vmmvmvm )( 212211

.20105.2

105

,)2500(0)50)(1000(

:1500,1000ifE.g.,

3

4

21

smkg

smkgv

vkgsmkg

kgmkgm

f

f

fi vmmvm )(0 2111

Perfectly Inelastic Collisions:

• What amount of KE lost during collision?

Jsmkg

vmvmKE iibefore

62

222

211

1025.1)50)(1000(2

12

1

2

1

Jsmkg

vmmKE fafter

62

221

1050.0)20)(2500(2

1

)(2

1

JKElost61075.0

lost in heat/”gluing”/sound/…

More Types of Collisions

• Elastic collisions– both momentum and kinetic energy are conserved

• Actual collisions– Most collisions fall between elastic and perfectly

inelastic collisions

More About Elastic Collisions

• Both momentum and kinetic energy are conserved• Typically have two unknowns

• Solve the equations simultaneously

222

211

222

211

22112211

2

1

2

1

2

1

2

1ffii

ffii

vmvmvmvm

vmvmvmvm

Elastic Collisions:

• Using previous example (but elastic collision is assumed)

smkg

smkgsmkg

vmvmP iibefore

4

2111

100.2

)20)(1500()50)(1000(

smv

smv

f

f

1.31

7.26

2

1

For perfectly elastic collision:

J

JJ

vmvmKE iibefore

6

56

222

211

1055.1

1031025.1

2

1

2

1

222

211

6

22114

2

1

2

11055.1

100.2

ff

ff

vmvmJ

vmvmsmkg

Problem Solving for One -Dimensional Collisions

• Set up a coordinate axis and define the velocities with respect to this axis– It is convenient to make your axis coincide with one

of the initial velocities

• In your sketch, draw all the velocity vectors with labels including all the given information

Sketches for Collision Problems

• Draw “before” and “after” sketches

• Label each object – include the direction of

velocity– keep track of subscripts

Sketches for Perfectly Inelastic Collisions• The objects stick

together• Include all the velocity

directions• The “after” collision

combines the masses

Problem Solving for One-Dimensional Collisions, cont.

• Write the expressions for the momentum of each object before and after the collision– Remember to include the appropriate signs

• Write an expression for the total momentum before and after the collision– Remember the momentum of the system is

what is conserved

Problem Solving for One-Dimensional Collisions, final

• If the collision is inelastic, solve the momentum equation for the unknown– Remember, KE is not conserved

• If the collision is elastic, you can use the KE equation to solve for two unknowns

Glancing Collisions

• For a general collision of two objects in three-dimensional space, the conservation of momentum principle implies that the total momentum of the system in each direction is conserved

–

– Use subscripts for identifying the object, initial and final, and components

fyfyiyiy

fxfxixix

vmvmvmvm

andvmvmvmvm

22112211

22112211

Glancing Collisions

• The “after” velocities have x and y components

• Momentum is conserved in the x direction and in the y direction

• Apply separately to each direction

Problem Solving for Two-Dimensional Collisions

• Set up coordinate axes and define your velocities with respect to these axes– It is convenient to choose the x axis to coincide with

one of the initial velocities

• In your sketch, draw and label all the velocities and include all the given information

Problem Solving for Two-Dimensional Collisions, cont

• Write expressions for the x and y components of the momentum of each object before and after the collision

• Write expressions for the total momentum before and after the collision in the x-direction– Repeat for the y-direction

Problem Solving for Two-Dimensional Collisions, final

• Solve for the unknown quantities– If the collision is inelastic, additional information is

probably required– If the collision is perfectly inelastic, the final

velocities of the two objects is the same– If the collision is elastic, use the KE equations to

help solve for the unknowns

Contoh masalah

• Seorang astronout dengan massa 80Kg melemparkan kunci pas dengan kecepatan 3 m/s. Tentukan kecepatan astronout yang terpental setelah melemparkan kunci pas. Sebarapa jauhkah astronout bersangkutan terpisahd ari pesawat dalam selang 1 jam ?

Contoh masalah 2

• Sebuah mobil dengan massa 1300kg bertabrakan dengan sebuah truk bermuatan dengan massa 15000kg di perempatan jalan. Setelah tubrukan mobil dan truk menjadi satu dan bergeser sejauh 5 m membentuk sudut 30O terhadap arah gerak mobil. Jika koefisien gesekan statiknya sebesar 0.7, berakah kecepatan awal masing-masing kendaraan ?

Mobil

Truk

30O

Rocket Propulsion• The operation of a rocket depends on the law of

conservation of momentum as applied to a system, where the system is the rocket plus its ejected fuel– This is different than propulsion on the earth where

two objects exert forces on each other• road on car• train on track

Rocket Propulsion, cont.

• The rocket is accelerated as a result of the thrust of the exhaust gases

• This represents the inverse of an inelastic collision– Momentum is conserved– Kinetic Energy is increased (at the expense of the

stored energy of the rocket fuel)

Rocket Propulsion

• The initial mass of the rocket is M + Δm– M is the mass of the rocket– m is the mass of the fuel

• The initial velocity of the rocket is v

Rocket Propulsion

• The rocket’s mass is M• The mass of the fuel, Δm, has been ejected• The rocket’s speed has increased to v + Δv

Thrust of a Rocket

• The thrust is the force exerted on the rocket by the ejected exhaust gases

• The instantaneous thrust is given by

– The thrust increases as the exhaust speed increases and as the burn rate (ΔM/Δt) increases

t

Mv

t

vMMa e