fap0015 ch07 momentum

8/2/2019 FAP0015 Ch07 Momentum

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Conservation of Momentum

Collisions: elastic & inelastic

Centre of mass

22/01/2010 09:00Ch7 - Linear Momentum 1


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Lesson OutcomesAt the end of the lesson, students should be able to:

. .

2. statethe law of the conservation of momentum.

. s e erences e ween e as c an ne as c co s ons.

4. solve problems involvingcollisions usingthe law of.

5. define and determine impulse and impulsive force.

. .



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Linear Momentum

When a particle with mass, m, moves with velocity v,

we define its momentum, p, as a product of its mass

and velocity.

entumlinear mommvp =

p is a vector quantity in the same direction as

velocity, v.

The unit of its magnitude in SI is kg m/s.



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Linear Momentum If the particle has velocity components vx, vy, vz., its

momentum com onents are

.zzyyxx mv, pmv, pmvp ===

Differentiating momentum with respect to time gives

dt

vd

dt

dP cm= cmcm Ma

dt

vM == = extF

The linear momentum ofan isolated system is conserved.



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Isolated System

By a system, we simply mean a set of objects that

.

For any system, the forces that various particles exert on

Forces exerted of any part of a system by some agency

A system with no external forces is called an isolated

s stem. F = 0



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We define the total momentumP of the system as a vector sumof momenta of the individual bodies.

=

The total (net) momentum P of any n-particles is equal to the

.

vector sum ofpn of individual particle, is

=

nnvm......vmvm +++= 2211

n

cmMv=

where vcm is the velocity of the centre of mass of the system of particles.



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We are familiar with collisions between billiard balls,

between a tennis ball and a racquet, between football players,

between cars on the highways and many many many more...



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Collisions

Collision can be defined as any strong interaction

.

(automotive disaster).

,

collisions conserve kinetic energy as well.

e resu o a co s on s cons ra ne y e aws o

conservation of energy and conservation of momentum.

o s ons a nto t ree categor es:



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Collisions

Elastic collisions, which conserve the kinetic energy,

example a collision between two hard steel balls or billiard

a s.

Inelastic collisions, which do not conserve the kinetic

energy ss pa e . ome s os o ea , soun energy

and forth. Example, two lumps of clay colliding.

,

together after collision. In such collisions the KE loss is

maximum. Cars colliding, spitball striking the window

shade and so on are examples.



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An Elast ic Col l ision in 1-DCart



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Elastic Collisions

The basic equations representing conservation of

Initial momentumPi = Final momentumPf

ffii vmvmvmvm 22112211 +=+

Initial kinetic energyKEi = Final kinetic energyKEf

222112222112 ffii vmvmvmvm +=+



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Special Case of Elastic Collision:

n w n y

v1i

m1

Ifv2i= 0, conservation of momentum and kinetic energy:

222

ffi 221111 ...(1)

221111 ffi



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Solving equations (1) & (2) for v1fand v2f:

(3)111112

1

2

11

2

22 +== )v)(vv(vm)v(vmvm fififif

( )411122 = vvmvm fif

Eq (3)/Eq (4): 5112 += fif vvv

( )6111112 =+ fifi vvmvvm

( ) ( ) if vmmvmm 121121 =+



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(7)vv 121

1 ,mm

if

=

21 mm

(8)vv 121

12 ,

mm

if

+

=

Some interestin eneral conclusions from these formulas.



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=

mm 21 if

mm1

21

1

+

if v

mm

v 121

12

+

=

In this case v1 = 0 and v2 = v1i, the colliding object stops, and

the target one moves with same speed and direction.



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If m1 m2:

vmm

v 21

=mm

21

+

m2if v

mmv 1

21

2+

=

In this case the colliding object continues in the same direction after the

impact but with reduced speed while the target object moves ahead of at

.If m1 >> m2 the colliding object losses little speed while the target oneis given speed nearly twice v1i. (v2 =2v1i)



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If m1 m2:

mm 21 =

mm 21 +

if v

mm

mv 1

21

12

+=

In this case v1fis opposite to v1i, The lighter object bounces off the

.Since m2 is virtually infinite compared with m1, v1f = - v1i



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Conce tual Question

.

Do the velocities of these objectsnecessaril have a the same directionand (b) the same magnitude? Give your

reasoning in each case.



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REASONING AND SOLUTION

Yes. Momentum is a vector, and the two objects have the same

momentum. This means that the direction of each objects

momentum is the same. Momentum is mass times velocity, and the

direction of the momentum is the same as the direction of the

velocit . Thus the velocit directions must be the same.

No. Momentum is mass times velocity. The fact that the objects

have the same momentum means that the product of the mass andthe magnitude of the velocity is the same for each. Thus, the

magnitude of the velocity of one object can be smaller, for

exam le as lon as the mass of that ob ect is ro ortionall reater

to keep the product of mass and velocity unchanged.



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Can a single object have kinetic energy but nomomentum?

Can a system of two or more objects have a total

kinetic energy that is not zero but a total momentum

Account for your answer.



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REA I A D L TI If a single object has KE, it must have a velocity; therefore, it

.

In a system of two or more objects, the individual objects couldhave linear momentum that cancel each other. In this case, thelinear momentum of the system would be zero.

The kinetic energies of the objects, however, are scalar quantities,

objects would necessarily be nonzero.

Therefore, it is possible for a system of two or more objects to

have a total inetic energy that is not zero but a total momentumthat is zero.



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Inelastic Collisions

In an inelastic collisions the momentum of the system is conserved

if =

if KK But its kinetic energy is not conserved.

When objects stick together after colliding, the collision is

com letel inelastic and the maximum amount of KE is lost.

Consider a system of two cars of mass m1 and m2 on smooth level

track. One at rest while the other moves toward it with a speed v1i,

what is the speed of the cars after collisions if they are stucktogether after the collision?



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(Completely) Inelastic Collisions



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(Completely) Inelastic Collisions

Initial momentumPi = Final momentumPf

( ) fii vmmvmvm 212211 +=+

2211

mmvmvmv iif

++=Thus 0if 2

21

11=+=

ii v

mmvm

KE 2121

i imv=2

2211121 +== vmvm ii 21

22 +mm

( )

2

22111vmvm ii += 0if

2

1

2

11 == vvm i

Initial kinetic energyKEi Final kinetic energyKEf

21 mm +21

+mm



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Let the two cars be identical, m1 = m2 ,

KE 2121

i imv=

w at s t e c ange n net c energy a ter t e co s on

( ) ( ) KE21

221121212

2121f +

++=+= mm

vmvmmmvmm iif

( ) ( )21

1i121

21

2i21i121

mm

vm

mm

vmvm

+=

+

+= since v2i = 0

since m1 = m2 , i212

1141

1

11

21

f KEmm2

mKE === i

i vv

i21

ii21

if KEKEKEKEKEKEKE,inChange ===



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Example A 1200 kg car moving at 2.5 m/s is struck in the rear by a 2600 kg

truck moving at 6.2 m/s. If the vehicles stick together after the

,

the change in kinetic energy? (Assume that external forces may beignored).

Solution: The momentum conserved for inelastic collision)

f1i22i11fi vmmvmvm 2+=+==

m/s05)kg26001200(

..vf .=+=



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Example (Application)

In a ballistic pendulum, an

object of mass m is fired

the bob of a pendulum.The bob has a mass M, and

negligible mass. After the

collision, the object and

swing through an arc,

eventually gaining a height

.

terms ofm, M, v0 and g.



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After collisions, the momentum is conserved, but KE is not.

fvmmv +=0 The Pi = Pf

0vmMvf

+= The speed after the collision is,



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mm

2

111

The KE after the collision is

+= ++=+= mMmvv

Mmmvm ff 00

222

the PE at the height h

=

( ) ghmMm

mv +=

20

1

Solving for h is

+=

gv

mMmh

2

20

2

ghmMmv 2or 0

+=



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Impulse-momentum Theorem

The impulse, J, of a force isthe product of the average force, Favand the time interval, t , during which the force acts:

t)uv(mtmatFJ === if pp =

Impulse equalsmomentum change, it is a vector quantity

- . .

Both mass and velocity play a role in how object responds

,

linear momentum.



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Impulse

,

====f f

if JFdtppmvmvdp if

The integral of the force over the time is called the impulse, J,(kg m/s or Ns).

e orce s ca e mpu s ve orce.

Valid for a very large force in a very short time interval& essentially

Large force short time = Small force long time. Application: Safety features for cars (or bicycle helmets). Crumple

zones in cars and the padding in helmets are all there to increase thetime over which a collision will take place, thus reducing the force


.


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The collision time between a bat and a ball is very

, .strikes the ball, the magnitude of the force excreted onthe ball rises to a maximum value and then returns to

zero when the ball leaves the bat.

Time interval is t,

the magnitude of theaverage force is Fav



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Conceptual ProblemA stunt person jumps from the roof of a tall building, but no injury occurs

because the person lands on a large, air-filled bag. Which one of the

o ow ng es escr es w y no n ury occurs

(a) The bag provides the necessary force to stop him.

e ag re uces e mpu se o e person.

(c) The bag increases the amount of time the force acts on the person and

(d) The bag decreases the amount of time during which the momentum is

changing and reduces the average force on the person.

(e) The bag increases the amount of time during which the momentum is

changing and reduces the average force on the person.



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Problem

A 220-g ball falls vertically downward, hitting the floor with a

. . .

(a) Find the magnitude of the change in the balls momentum.

(b) Find the change in the magnitude of the balls momentum.

(c) Which of the two quantities calculated in parts (a) and (b) ismore directly related to the net force acting on the ball during its

collision with the floor? Explain.



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Center of mass (CM)

Observations indicate that even if a body rotates, or there are

several bodies that move relative to one another, there is one

-

the centre of mass. CM of a s stem is the oint where the s stem can be balanced in

a uniform gravitational field.

Consider the extended body made up of two tiny particles m1 &m2 , the position of CM,

xmxmxmxmX 22112211

+=

+=

mm 21 + the position of CM lies on the line joining m1 & m2.



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Center of Mass

Suppose the coordinates of m1 be (x1, y1) of m2 be (x2, y2),

etc we define center of mass of the s stem as the oint

having coordinates (xcm, ycm) given by

m 1 2m 3m0 x

xmxmx

n

i ii

n

i ii

CM

== == 11mi i=1



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In 3-D, the coordinates of the C of M are:

xmx

ii

CM

=

ym ii=

zmz

ii=

M M

The osition vector of center of mass can be ex ressed in

terms of the position vectorsr1, r2,

iirm

=

+++=

i

icm

mmmmr

.....

...

321

332211

In statistical language the center of mass is a mass-weightedaverage position of the particles.



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If m1= m2 = m then XCM=( x1 + x2 )/2 (midway) Ifm1 > m2 then CM is closer to the larger mass.

If all the masses concentrated atx so that m = 0

XCM= m2x2/ m2 = x2 as expected



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Example

A square uniform raft, 18 m by 18 m, of mass 6800 kg,. ,

kg, occupy its NE, SE, and SW corners, determine thecm of the loaded ferr boat.

( )( ) ( )( ) ( )( )1200 kg 9 m 1200 kg 9 m 1200 kg 9 m1.04mCMx

+ +

= =

( )( ) ( )( ) ( )( )1200 kg 9 m 1200 kg 9 m 1200 kg 9 m+ +

( ).

3 1200 kg 6800 kgCM

y = =

+



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Conclusions Impulse == tFJ

mvp = Linear momentum The total momentum P of a system remains constant,whenever the vector sum of the external forces of it is zero.

,0)( 21 =+= FFdt

.

Inelastic collisions conserve the momentum and do notconserve the kinetic energy (dissipated).

CM of a system is the point where the system can be balancedin a uniform gravitational field.



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Review

Complete the following statement: A collision is elasticif

(a) the final velocities are zero.

.

(c) the objects stick together.

(d) the total kinetic energy is conserved.

.



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Choose the Correct Answer

Which one of the following is characteristic of an

(a) Total mass is not conserved.

ne c energy s no conserve .

(c) Total energy is not conserved.

(d) The change in momentum is less than the total

impulse.

(e) Linear momentum is not conserved.



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Exercise An object of mass 3m, initially at rest, explodes breaking into two

fragments of mass m and 2m, respectively. Which one of thefollowin statements concernin the fra ments after the ex losionis true?

(a) They may fly off at right angles.

ey may y o n e same rec on.

(c) The smaller fragment will have twice the speed of the larger

fragment. (d) The larger fragment will have twice the speed of the smaller

fragment.

larger fragment.


fap0015 ch07 momentum

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