linear momentum - uml.edufaculty.uml.edu/andriy_danylov/teaching/documents/... · 2013. 10. 30. ·...

Department of Physics and Applied Physics95.141, Fall 2013, Lecture 15

Course website:http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI

Lecture Capture: http://echo360.uml.edu/danylov2013/physics1fall.html

Lecture 15

Chapter 9

Linear Momentum

10.30.2013Physics I


Chapter 9

Momentum Conservation of momentum Impulse Collisions (Elastic & Inelastic)

Outline


Exam II Info

Exam II Mon Nov 4, 9:00-9:50am, OH 150.Exam II covers Chapters 5-8

Same format as Exam IPrior Examples of Exam II posted

Ch. 5: Using Newton’s Laws, Friction, Uniform Circular motionCh. 6: Universal Law of Gravitation, Kepler’s Laws

Ch. 7: Work & EnergyCh. 8: Conservation of Energy

Exam Review Session Thursday 6-8pm, Ball 210


Linear Momentum

Linear momentum is defined as the product of an object’s mass and velocity:

vmp

Units of momentum: smkg

The greater the linear momentum of a body , the greater its tendency to continue in motion.

v

m

An iron shot (m larger) is harder to stop than a baseball (m small) of the same velocity.

Momentum is a VECTOR !


Force and Momentum (Newton’s 2nd law)

F ma d

dt(mv) m d

vdt

dp

dt

Let’s rewrite Newton’s 2nd law in terms of momentum:

The rate of change of momentum is equal to the net force

dtpdF

So, a force is required to change momentum of an object.


How to find an average force?

Faverage

p

tpf

pit

This can be used to get average force from momentum change:

netF

tii vmp

ff vmp

dtpdF

So, let’s rewrite N. 2nd law for an average force:

Let’s look at an example:


Example: finding an average forceThe speed of a fastball is about 40 m/s, and the speed of the ball coming off of player’s bat for a home run is about 54 m/s. The ball (0.145kg) is in contact with the bat for 1ms. What is the average Force exerted by the player?

NFaverage 300,136

tpp

F ifaverage

ssmkgFaverage 001.0

/]4054)[145.0(

tvvm if

))((

in the direction of xorv f

x

Pay attention to directions!!!!!!!!

ConcepTest 1 Two Boxes/Momentum

F F light heavy

We know:

In this case F and t are the same for both boxes!Both boxes will have the same final momentum.

A) the heavier one

B) the lighter one

C) both the same

Two boxes, one heavier than the other, are initially at rest on a horizontal frictionless surface. The same constant force F acts on each one for exactly 1 second. Which box has more momentum after the force acts ?

tpp

tpF ifav

tFp avf

0ip

In the previous question,

which box has the larger

velocity after the force acts?

A) the heavier one

B) the lighter one

C) both the same

ConcepTest 2 Two Boxes/velocity

lfh mvpMv

hl vvthenmMSince ,


Conservation of Momentum

If no net external force acts on a system, its momentum is conserved.

0 externalFIf

dtpdF

From Newton’s 2nd law:

0, dtpdthen

thus, constp

AAvm

BBvm

AAvm

BBvm

Pinitial mA

vA mBvB

Pfinal mA

vA' mB

vB'

mAvA mB

vB mAvA

' mBvB

'Pinitial

Pfinal


Why is Momentum conserved?

gm 1 gm

2

1NF

2NF

mgs are canceled by normal forces, so net external force is zero,

and the momentum is conserved

gm 1

gm 2

2NF

The net external force is m1g, and the momentum is NOT conserved

Isolated system is a system on which no external forces act. There are only internal forces acting between objects.

this system (two balls) is isolated this system (two balls) is NOT isolated

The total momentum of an isolated system of objects is conserved


Internal forces of an isolated systemInternal forces of an isolated system.

A acts on B:

B acts on A:

Forces equal and opposite (Newton’s 3rd Law).

Thus, they all cancel each other. That’s why they cannot ruin conservation of momentum.(It is not a proof)

ABF

BAF

BAAB FF

ABF

BAF

During collision


Impulse

F d

pdt

Impulse= change in momentum

JdtFf

i

t

t

F dt

ti

t f dppipf pf pi p

Define Impulse as:

pddtF

From Newton’s 2nd law:

So, Impulse= area under F-vs-t curveJ

pJ

Integrate it:

During a collision, objects are deformed because of the large forces involved . How to relate those forces with a change in momentum?

Force exerted on one of the balls

Befo

re c

ollis

ion

Afte

r col

lisio

n

fp

ip


Impulse/Average force

pdtFtFf

i

t

tavg

J

The exact variation of F with time is very often not known. So, it is easier to find an average force.

JJsamethe

Having a certain ∆p, a cat by bending its lags tries to increase ∆t (impact time), so that an impact force would be reduced. (intuitive knowledge of Physics )

How to avoid broken legs

ptFavg


Example: Tennis ball/impulseThe force exerted by a tennis racket on the ball (mass 56 g)during a serve ( )can be approximated by the F vs time plot below.What is the impulse on the ball? What is the speed of the serve?

m/s 71kg 056.0sN 4

fv

Forc

e (k

N)

Time (ms) 10

2Area under force-time curve is an impulse:

JdtFf

i

t

t

pf pi

0

sNmskNAreaJ 4)2

22(2

ff mvpJ

0

mJv f

0iv


Momentum is conserved in any two-body collision(since there is no net external force)

Different types of collisions

Is mechanical energy conserved in these collisions?

Mech. energy is conserved Mech. energy is NOT conserved

Metal balls get deformed and restored Cars get deformed and not restored. Some Mech. Energy is spent on deformation.

Inelastic collisionElastic collision


1-D Elastic Collisions


Elastic Collision Math (1D)

Relative velocities switch signs in the collision

mAvA mBvB mA vA mB vB 12 mAvA2 12 mBvB

2 12 mA vA2 12 mB vB

2

mAvA mA vA mB vB mBvB mAvA2 mA vA

2 mB vB2 mBvB

2

mA (vA vA ) mB ( vB vB ) mA (vA2 v

A

2 ) mB ( vB2 vB

2 )

mA (vA vA )(vA vA ) mB ( vB vB )( vB vB )

vA vA vB vB

)( BABA vvvv

Conservation of momentum Conservation of mech. energy

AAvm

BBvm

AAvm

BBvm


Elastic Collision Math (1D)

mAvA mBvB mA vA mB vB12 mAvA

2 12 mBvB2 12 mA vA

2 12 mB vB2

)( BABA vvvv

Conservation of momentumConservation of mechanical energy

mAvA mBvB mA vA mB vB Conservation of momentumConservation of mechanical energy

So, instead of the 1st set of “crazy” equations, we can use the 2nd one which is easier (both are linear)

1st

2nd


Example: Ballistic Pendulum

A device used to measure the speed of a bullet.

hvom M v1M+m


Example: Ballistic Pendulum (cont.)

the speed of a bullet

ForBullet mass 10 g Block mass is 3 kgBlock swings up to a height of 5 cm

sm

ov 298


A Different Ballistic “Pendulum”

Bullet mass 30 g Block mass is 5 kgSpring compresses by 12 cmSpring constant k = 300 N/mBullet velocity before collision?

m

mv0 (M m)v1

v0 M m

mv1

12

(M m)v12 1

2kx2

v1 xk

M m

M

v1 0.93m svo 155.4 m s

A device used to measure the speed of a bullet.


Thank youSee you on Monday

linear momentum - uml.edufaculty.uml.edu/andriy_danylov/teaching/documents/... · 2013. 10. 30. ·...

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