Linear Momentum

MOMENTUM

Momentum is a commonly used term in sports.

Momentum is a physics term; it refers to the quantity

of motion that an object has.

LINEAR MOMENTUM

COURSE OUTLINE

A. Introduction

B. Kinematics

C. Dynamics

D. Newton's Law of Universal Gravitation

E. Energy

F. Linear Momentum

G. Fluid Mechanics

H. Thermodynamics

Outline

Linear Momentum

1. Linear Momentum

2. Momentum and Newton's Second Law

3. Impulse of a Force

4. Impulse Momentum Relations

5. Conservation of Momentum

Collisions

6. Overview ~ Collisions

7. Elastic and Inelastic Collisions

Vector quantity, the direction of the momentum is the

same as the velocitys

Inertia in motion

Applies to two-dimensional motion as well

yyxx mvpandmvp

vmp

Size of momentum: depends upon mass

depends upon velocity

LINEAR MOMENTUM

LINEAR MOMENTUM

Physical Properties

Symbol: p

Type: Derived, Vector

Dimension: [M*L/T]

SI unit: kg m/s

p = m v

Can be thought of as the effort you need to stop an object from moving.

Determined by two factors:

1. The objects inertia (mass)

2. The objects velocity

LINEAR MOMENTUM

For example, a heavy truck has more momentum than a light car travelling at the same speed.

It takes a greater force to stop the truck in a given time than it does to stop the car

MOMENTUM and NEWTONS 2nd LAW OF MOTION

Newton's Second Law can be written in terms of the momentum of a particle.

but p = mv, so much p = mv

Thus the net force acting on a particle equals the time rate change of the particle's linear momentum

This is how Newton originally stated his second law!

mutatio motus change of motion caused by the force impressed

MOMENTUM and NEWTONS 2nd LAW OF MOTION

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 Newtons second law

(F t) 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

:

)(

Physical Properties:

Symbol: I

Type: Derived, Vector Quantity

Formula:

I = F t = F (tf t

i)

Dimension [F*T]; SI Units: N*s (Newton*second) 1 N*s = 1 kg m/s2 *s= 1 kg m/s

Impulse

tFp net

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 i

t

impulse F t area under F t curve

If force is constant: impulse = F t

IMPULSE-MOMENTUM RELATIONS

I net = F t

Inet

= p = pf p

i

The average force for the time interval tf t

iis defined

as F

av= I / t

The average force is the constant force that gives the same impulse as the actual force in the time interval t.

This time is often estimated using the distance travelled by one of the objects during the collision.

Minimizing the force of impact

Impulse is associated with the forces of interaction during collisions.

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 time of the collision and absorbs some of the energy from the body

Problem:

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 ball

b) Assuming club in contact

with ball for 0.5 ms, find

average force acting on golf

ball

Problem:

Given:

mass: m=50 g

= 0.050 kg

velocity: 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 Newtons 3rd law, that is also a reaction force to club hitting the ball:

iiff

ifif

R

VMvmVMvm

orVMVMvmvm

ortFtF

,

,of club

CONSERVATION OF MOMENTUM

The total momentum P of a system of particles is the sum of the momenta of the individual particles.

P = miv

i= p

i

According to Newton's Second Law,

Fext

= Fnet,ext

= P/t = mv/t = ma

Linear Momentum

Pinitial

= Pfinal

More applicable than the law of conservation of mechanical energy.

Conservation of Momentum

REASON: Although internal forces exerted by one particle in a system on another are often NOT CONSERVATIVE. Internal forces can change the total mechanical energy of the system but they DON'T affect the total 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 which a collision occurs is conserved (regardless of the

nature of the forces between the objects)

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


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

You can use a golf club for all kinds of non-golfy purposes -- walking stick,

fishing rod, club, to name three. And now we can add to that list --

firestarter.

Over the weekend, a golfer's routine swing in the rough at the Shady Canyon

Golf Course in Irvine, Calif., struck a rock. Not so different from the way

you play, right? Only this time, the impact caused a spark, and the spark set

off a blaze that eventually covered 25 acres (101171.41056 Square Meters),

according to the Steven Buck, General Manager of Shady Canyon Golf

Course, and required the efforts of 150 Orange County firefighters, writes

the Associated Press.

Wow. And I felt bad the time I shanked a ball through the window of a house

too close to the fairway. That was nothing compared to this!

The golfer's name is being withheld, which is probably for the best, and no

charges are going to be filed. Fortunately, it all could have been much worse.

As it was, the blaze required both helicopters and on-the-ground crews.

In a collision, two objects approach and interact strongly

for a very short time.

During this brief time of collision,

F ext

ELASTIC COLLISION andINELASTIC COLLISION

When the total kinetic energy of the objects is the same after collision as before the collision is called an elastic collision

When the total kinetic energy of the objects is not the same, it is termed an inelastic collision.

NOTE: Actual collisions

Most collisions fall between elastic and perfectly inelastic collisions

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

2

22

2

11

1025.1)50)(1000(2

1

2

1

2

1

Jsmkg

vmmKE fafter

62

2

21

1050.0)20)(2500(2

1

)(2

1

JKElost61075.0

lost in heat/gluing/sound/

Elastic Collisions

Both momentum and kinetic energy are conserved

Typically have two unknowns

Solve the equations simultaneously

2

22

2

11

2

22

2

11

22112211

2

1

2

1

2

1

2

1ffii

ffii

vmvmvmvm

vmvmvmvm

Problem Solving Tips:

If the collision is inelastic, KE is not conserved

If the collision is elastic, KE is conserved

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

Two-dimensional 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

vmvmvmvm

22112211

22112211 and

ffii vmvmvmvm 22112211

Example:

What would happen afterthe collision?

Stationary

It is also possible for two bodies to undergo scattering

Example:

What would happen after the collision?

Stationary

It is also possible for two bodies to undergo scattering

Assume: m1=m2 and v1i=5 m/s

For this problem: assume that q = f = 60

Example:

Given:

masses: m1=m2velocity: v1i=5 m/s

v2i=0 m/s

angles: q = f = 60

Find:

v1f = ?

v2f = ?

Use momentum conservation in each

direction (x and y):

ff

ff

yiffyf

vv

mmvv

pvmvmp

21

2121

2211

as,60sin60sin

060sin60sin

smvv

smvv

smmpvmvmp

ff

ff

xiffxf

5

55.05.0

560cos60cos

21

21

12211

Two cars collide at a intersection. Car 1 has a

mass of 1200 kg and is moving at a velocity of

95.0 km/hr due east and car 2 has mass of

1400 kg and is moving at a velocity of

100km/hr due north. The cars stick together

and move off as one at an angle (wrt x-axis). Find (a) the angle and (b) the final velocity of the combined cars.

Example:

Ballistic Pendulum

In a feat of public marksmanship, Juzzel fires a bullet into a hanging

target. The target, with bullet embedded, swings upward.

Noting the height reached at the top of the swing, he immediately

inform the crowd of the bullet's speed. For arbitrary masses: m1

(bullet), m2

(hanging target), and h (height, top of the swing), how

did he calculate the bullet's speed? (See Figure above)

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Linear Momentum