1 LINEAR MOMENTUM.pdf

MOMENTUM…

Momentum is a commonly used term in sports.

COURSE OUTLINE

A. Introduction

B. Kinematics

C. Dynamics

D. Newton's Law of Universal Gravitation

E. Energy

F. Linear Momentum

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

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

 Inertia in motion

 Applies to two-dimensional motion as well

y y

x

x

m v

and

p

m v

p





v

m

p



Size of momentum: depends upon mass depends upon velocity

LINEAR MOMENTUM



Physical Properties



Symbol:

p



Type: Derived, Vector

Dimension: [M*L/T]

SI unit: kg m/s

Can be thought of as the effort you need to stop an

object from moving.

Determined by two factors:

1. The object’s inertia (mass) 2. The object’s velocity

LINEAR MOMENTUM

For example, a heavy truck has more momentum

than a light car travelling at the same speed.

MOMENTUM and NEWTON’S 2

LAW OF

MOTION

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

 but p = mv, so much Δp = mΔv

This is how Newton originally stated his second

law!

mutatio motus –”change of motion” caused by the

force impressed

MOMENTUM and NEWTON’S 2

LAW OF

QUIZ

 What is the momentum of a 2 kg object moving with a velocity of 10.0 m/s to the right?

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

t F p or a m t v v m t p F net i f

net    

Impulse

For a specified object of mass m,

Change in Momentum

Means

CHANGE IN VELOCITY!

Impulse

Change in velocity Means

Net Force acting on the object is NOT ZERO

Impulse is related with a FORCE causing the object to

Physical Properties:



Symbol:

I



Type: Derived, Vector Quantity

Formula:



I

=

F

Δt =

F

(t

– t

)



Dimension [F*T]; SI Units: N*s (Newton*second)

 1 N*s = 1 kg m/s2 *s= 1 kg m/s

Impulse

t F

p I

Impulse

 Impulse then can be expressed as: I or J

I=

=

ΣF

Δt

or simply

I

=

F

t = Δ

p

Restating Newton’s 2

Law of Motion

 A non – zero net force applied on the object the changes the object’s momentum.

Question:

A.) Does a moving object have impulse?

IMPULSE-MOMENTUM RELATIONS

I

= F

Δt

I

=

Δp = p

– p



The average force for the time interval t

– t

is defined as

F

= I / Δt



The average force is the constant force that gives the

same impulse as the actual force in the time interval Δt.



This time is often estimated using the distance travelled

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

Impulse is change in momentum

Case 1: Increasing momentum:

If you wish to increase the momentum of an object as much as possible, you not only apply the greatest force you can, you also extend the time of application as much as possible.

Net force

Case 2: Decreasing momentum

 You can change the momentum of an object in two ways

A.) Increasing the coOOOOOntact Time

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







Problem:

A 50-g golf ball at rest is hit by the 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

Problem:

Given:

mass: m=50 g = 0.050 kg velocity: v=50 m/s

Find: impulse=? F_average=?

1. Use impulse-momentum relation:

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

   s m kg s m kg mv mv p

impulse _{f i}

Conservation of Momentum

Note:

according to Newton’s 3

law, that is also a

reaction force to club hitting the ball:





V

M

v

m

V

M

v

m

or

V

M

V

M

v

m

v

m

or

t

F

t

F



























,

,

of club



The total momentum

p

of a system of particles is

the sum of the momenta of the individual

particles.

P = Σ m

v

= Σ p



According to Newton's Second Law,

Σ F

= F

=

ΔP

/Δt =

mΔv

/Δt = ma

P

initial

= P

final

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

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



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

f f

i

i

m

v

m

v

m

v

v

m







 In a collision, two objects approach and interact strongly for a very short time.

 During this brief time of collision,

F ext << F interaction between two objects

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

energy

lost





i

K E

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!

ELASTIC COLLISION and INELASTIC COLLISION

Elastic collision

 _{total kinetic energy of the objects is the same after collision as}

before the collision

Inelastic collision

 total kinetic energy of the objects is not the same

after collision as before the collision

An extreme case is the perfectly inelastic collision, in which all of the kinetic energy relative to the centre of mass is converted to thermal or internal energy of the system, and the two objects STICK TOGETHER!!!

NOTE: Actual collisions

Perfectly Inelastic Collisions:

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

 Suppose, for example, v_2i=0. Conservation of momentum becomes

f i

i

m

v

m

m

v

v

m





(



)

. 20 10 5 . 2 10 5 , ) 2500 ( 0 ) 50 )( 1000 ( : 1500 , 1000 if E.g., 3 4 2 1 s m kg s m kg v v kg s m kg kg m kg m f f          f i m m v

v

Perfectly Inelastic Collisions:

 What amount of KE lost during collision? J s m kg v m v m

KE_{before i i}

6 2 2 2 2 2 1 1 10 25 . 1 ) 50 )( 1000 ( 2 1 2 1 2 1      J s m kg v m m

KE_{after f}

6 2 2 2 1 10 50 . 0 ) 20 )( 2500 ( 2 1 ) ( 2 1      J K E_{lo st}  0.75106



Elastic Collisions

 Both momentum and kinetic energy are conserved

 Typically have two unknowns

 Solve the equations simultaneously

Example 1

Quiz:

1. A 2.0 N-force from a certain load stops a 2.0 kg-object initially moving at 10.0 m/s.

a. Compute for the impulse imparted to the object. b. What force was applied by the object on the load?

2. (True or False) The momentum of a system may be conserved even when mechanical energy is not.

4. Consider two less-than-desirable options. In the first you are driving 30 miles/hr and crash head-on into an identical car also going 30 miles/hr. In the second option you are driving 30 mph and crash head-on into a stationary brick wall. In neither case does your car bounce off the thing it hits, and the collision time is the same in both cases. Which of these two situations would result in the greatest impact force?

A. first B. second C. both

Example 2

Example 3

 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: m

1 (bullet),

m

2 (hanging target), and h

(height, top of the swing), how did he calculate the bullet's

Problem solving: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 fy fy iy iy fx fx ix ix v m v m v m v m v m v m v m v m 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 and       f f i

i

m

v

m

v

m

v

v

Example:

What would happen after the collision?

Stationary

Example:

What would happen after the collision?

Stationary

It is also possible for two bodies to undergo scattering Assume: m₁=m₂ and v_1i=5 m/s