vectors again. equations for motion along one dimension

Linear MomentumVectors again

Review Equations for Motion Along One Dimension

dt

dx

t

xv

t

xv

t

ave

0lim

dt

dv

t

va

t

va

t

ave

0lim

Review Motion Equations for Constant Acceleration

•1.

•2.

•3.

•4.

atvv 0

221

00 attvxx

20vv

vave

xavv 220

2

Review 3 Laws of Motion If in Equilibrium

If not in equilibrium Change in Motion is Due to Force

Force causes a change in acceleration

0F

maF

Work

Energy

Review

UW

FddFW

KWW

grav

Total

cos

2

2

2

1

2

1

kxU

mgyU

mvK

spring

grav

Law of conservation of energy

Power

efficiency

Review

NCspringgravspringgrav WUUKUUK 000

aveave Fvt

WP

in

out

P

Pe

If an 18 wheeler hits a car, what direction will the wreckage move?

What is the force between the 18 wheeler and the car?

Collisions

Forces

dt

vdmF

dt

vdmF

amF

Newtons 2nd law

Linear momentum

SI

Newton defined it as quantity of motion

Momentum

vmp

sN

s

mkg

dt

pdF

When an object collides with another, the forces on the object will momentarily spike before returning back to zero.

Impulse

Impulse

pJ

ptF

t

pF

dt

pdF

We now define impulse, J, as the change in momentum of a particle during a time interval

SI unit

Impulse

pJ

tFpJ

tFpJ

aveyyy

avexxx

sNs

mkg

A ball with a mass of 0.40 kg is thrown against a brick wall. It hits the wall moving horizontally to the left at 30 m/s and rebounds horizontally to the right at 20 m/s. (a) find the impulse of the net force on the ball during the collision with the wall. (b) If the ball is in contact with the wall for 0.010s, find the average horizontal force that the wall exerts on the ball during impact.

Example

Example

Nt

JF

tFJ

sNJ

vvmJ

vmvmJ

pJ

ave

ave

2000010.0

20

20

))30(20(4.0)( 0

0

If a particle A hits particle B

Conservation of Momentum

dt

pdF

dt

pdF

dt

pdF

ABonA

BAonB

If there are no external forces acting on the system


0

0

dt

pd

dt

pdFF

F

ABBonAAonB

Change in momentum over time is zero

The sum of momentums is constant


0)(

dt

ppd

dt

pd

dt

pd BAAB

constpp BA

If there are no external forces acting on a system, Total Momentum of a system conserved


constpp BA

BABA pppp

00

pp

0

Vector Addition

yx AAA cosAAx sinAAy

A marksman holds a rifle of mass 3.00 kg loosely such that it’ll recoil freely. He fires a bullet of mass 5.00g horizontally with velocity relative to the ground of 300 m/s. What is the recoil velocity of the rifle?

Example - Recoil

Example - Recoil

sm

R

sm

RRRR

sm

BBB

RBx

v

vvmp

vmp

ppp

5.0

5.1)3(

15)300)(05.0(

0

Two battling robots are on a frictionless surface. Robot A with mass 20 kg moves at 2.0 m/s parallel to the x axis. It collides with robot B, which has a mass of 12 kg. After the collision, robot A is moving at 1.0 m/s in a direction that makes an angle α=30o. What is the final velocity of robot B?

Example – 2D example

Example

sm

Bx

Bx

B

AxAAxABx

BxBAxAAxA

BxBAxAAxA

BxAxBxAx

xx

v

v

m

vmvmv

vmvmvm

vmvmvm

pppp

pp

89.112

30cos)1)(20()2(20

0

0

0

00

0

Example

sm

By

By

B

AyABy

ByBAyA

ByBAyA

ByAyByAy

yy

v

v

m

vmv

vmvm

vmvm

pppp

pp

83.012

30sin)1)(20(

0

00

0

Example

2489.1

83.0arctan

1.2

83.0

89.1

22

sm

B

ByBxB

sm

By

sm

Bx

v

vvv

v

v

Elastic Collisions – Collisions where the kinetic energies are conserved. When the particles are in contact, the energy is momentarily converted to elastic potential energy.

Conservation of Momentum and Energy

Inelastic Collisions – collisions where total kinetic energy after the collision is less than before the collision.

Completely Inelastic Collisions- When the two particles stick together after a collision.

Collisions can be partly inellastic

Conservation of Momentum and Energy

Collisions where two objects will impact each other, but the objects stick together and move as one after the collision.

Completely Inelastic Collisions

Momentum is still conserved Find v in terms of v0


vmvmvmvm

vvv

vmvmvmvm

pppp

pp

BABBAA

BA

BBAABBAA

BABA

00

00

00

0

Assume Particle B is initially at rest


BA

AA

BAAA

BABBAA

mm

vmv

vmvmvm

vmvmvmvm

0

0

00

Kinetic Energy


BA

A

BA

AA

BA

AABABA

AA

mm

m

K

K

mm

vmK

mm

vmmmvmmK

vmK

0

20

2

02

200

)(2

1

)(2

1)(

2

1

2

1

If B is at rest

At the intersection, a yellow subcompact car with mass travelling 950 kg east collides with a red pick up truck with mass 1900 kg travelling north. The two vehicles stick together and the wreckage travels 16.0 m/s 24o E of N. Calculate the speed of each of the vehicles. Assume frictionless.

Examples – Young and Freedman 8.37

Young and Freedman 8.37

B

yBAyB

A

xBAxA

y

x

yByAyBB

xBxAxAA

BABBAA

m

vmmv

m

vmmv

vv

vv

vmvmvm

vmvmvm

vmvmvmvm

)(

)(

24cos

24sin

0

0

0

0

00


sm

B

yBAyB

sm

A

xBAxA

y

x

m

vmmv

m

vmmv

vv

vv

9.21)(

5.19)(

6.1424cos1624cos

51.624sin1624sin

0

0

The ballistic pendulum is an apparatus to measure the speed of a fast moving projectile, such as a bullet. A bullet of mass 12g with velocity 380 m/s is fired into a large wooden block of mass 6.0 kg suspended by a chord of 70cm. (a) Find the height the block rises (b) the initial kinetic energy of the bullet (c) The kinetic energy of the bullet and block.

Problem – Ballistic Pendulum

Velocity after impact

Kinetic energy after impact


BA

AA

BAAA

BABBAA

mm

vmv

vmvmvm

vmvmvmvm

0

0

00

J

mm

vmK

BA

AA 73.1)(2

1 20

Kinetic energy after impact

Converted to potential at highest point


)(2

1 20

BA

AA

mm

vmK

gymm

mm

vm

gymmU

mm

vmK

BABA

AA

BA

BA

AA

)()(2

1

)(

)(2

1

20

20


my

y

mm

vm

gy

gymmmm

vm

BA

AA

BABA

AA

0293.0

)012.6)(8.9(2

))380(012.0(

)(2

1

)()(2

1

2

2

2

20

20

JK

K

mvK

Bullet

Bullet

BBullet

866

)380(012.02

12

1

2

2

Momentum and Energy are conserved Find v in terms of v0

Elastic Collisions

BBAABBAA

BABA

vmvmvmvm

pppp

pp

00

00

0

2220

20

00

0

2

1

2

1

2

1

2

1BBAABBAA

BABA

vmvmvmvm

KKKK

KK

If particle B is at rest

Elastic Collisions – One Dimension

BBAAAA

BABA

vmvmvm

pppp

pp

0

00

0

2220

00

0

2

1

2

1

2

1BBAAAA

BABA

vmvmvm

KKKK

KK



BBAAA

BBAAAA

BBAAAA

vmvvm

vmvmvm

vmvmvm

)( 0

0

0

2220

2220

2220

)(

2

1

2

1

2

12

1

2

1

2

1

BBAAA

BBAAAA

BBAAAA

vmvvm

vmvmvm

vmvmvm


Substitute back


BAA

BBAAA

BBAAA

vvv

vmvvm

vmvvm

0

2220

0

)(

)(

)(

)(

)()(

)()(

0

0

00

00

00

BA

ABAA

ABAABA

ABAAABAA

ABABAAAA

AABAAA

mm

vmmv

vmmvmm

vmvmvmvm

vmvmvmvm

vvmvvm



)(

2

)(

)(

)(

)(

0

00

0

0

BA

AAB

BBA

ABAA

BAA

BA

ABAA

mm

vmv

vmm

vmmv

vvv

mm

vmmv

If ma <<< mb

really small


)(

2

)(

)(

0

0

BA

AAB

BA

ABAA

mm

vmv

mm

vmmv

B

AA

BA

AAB

AB

ABA

m

vm

mm

vmv

vm

vmv

00

00

2

)(

2

If ma>>>mb

If ma=mb


00

00

22

AA

AAB

AA

AAA

vm

vmv

vm

vmv

00

0

2

2

0)(

)0(

AA

AAB

BA

AA

vm

vmv

mm

vv

In a game of billiards a player wishes to sink a target ball in the cornet pocket. If the angle to the corner pocket is 35o, at what angle is the cue ball deflected? (Assume frictionless)

Example

Mass is the same

Example

2220

20

00

BBAABBAA

BBAABBAA

vmvmvmvm

vmvmvmvm

2220

0

BAA

BAA

vvv

vvv

Example

55

3590

0cos

cos20

20

2

)()(222

0

20

0020

2220

0

BA

BA

BBAAA

BABAA

AAA

BAA

BAA

vv

vv

vvvvv

vvvvv

vvv

vvv

vvv

Two particles with masses m and 3m are moving towards each other along the x axis with the same initial speeds. Particle m is travelling towards the left and particle 3m is travelling towards the right. They undergo an elastic glancing collision such that particle m is moving downward after the collision at right angles from initial direction. (a) Find the final speeds of the two particles. (b) What is the angle θ at which particle 3m is scattered.

Problem – Serway 9-36

Elastic Collisions and relative velocity – One Dimensional

ABBA

BBAA

BBBAAA

BBBBAAAA

BBBAAA

BBBBAAAA

BBAABBAA

BBAABBAA

vvvv

vvvv

vvmvvm

vmvmvmvm

vvmvvm

vmvmvmvm

vmvmvmvm

vmvmvmvm

00

00

20

2220

20

2220

00

00

2220

20

00

)()(

)()(

In an elastic Collision, the relative velocities of the two objects have the same magnitude

Elastic Collisions and relative velocity

ABAB

ABEAEB

EAEBEAEB

EAEBEBEA

ABBA

vv

vvv

vvvv

vvvv

vvvv

|0|0

|||

|||0|0

|||0|0

00

)(

A 0.150 kg glider (puck on an air hockey table) is moving to the right with a speed of 0.80 m/s. It has a head-on collision with a 0.300 kg glider that is moving to the left with velocity 2.20 m/s. Find the final velocities of the two gliders. Assume elastic collision.


A bat strikes a 0.145kg baseball. Just before impact the ball is travelling horizontally to the right at 50.0 m/s and it leaves the bat travelling to the left at an angle of 30o above the horizontal with a speed of 65.0 m/s. Find the horizontal and vertical components of the average force on the ball if the ball and bat were in contact for 1.75 ms.

Problems- Young and Freedman 8.12

A 23 g bullet travelling at 230 m/s penetrates a 2.0kg block of wood and emerges cleanly at 170 m/s. If the wood is initially stationary on a frictionless surface, how fast does it move after the bullet emerges?

Giancoli 7-12

A 90.0 kg full back running east with a speed 5.0 m/s is tackled by a 95.0kg opponent running north at 3.00 m/s. If the collision is completely inelastic, (a) find the velocity of the players just after the tackle. (b) find the mechanical energy lost during the collision.

Serway 9.28

Giancoli 7-78 A 0.25kg skeet (clay target) is fired at an

angle of 30o to the horizon with a speed of 25 m/s. When it reaches its maximum height, it is hit from below by a 15g pellet traveling vertically upwards at 200 m/s. The pellet is embedded into the skeet. (a) how much higher does the skeet go up? (b) how much further does the skeet travel?

Prepare for pain

Objects approximated to be point particles Objects only undergo translational motion

Assumptions so far

Real objects also undergo rotational motion while undergoing translational motion.

But there is one point which will move as if subjected to the same net force.

We can treat the object as if all its mass was concentrated on a single point

Center of Mass

Set an arbitrary origin point

Center of mass is the mass weighted average of the particles

Center of Mass

i

iicm m

mr

mmm

mrmrmrr

...

...

321

332211

A simplified water molecule is shown. The separation between the H and O atoms is d=9.57 x10-11m. Each hydrogen atom has a mass of 1.0 u and the oxygen atom has a mass of 16.0 u. Find the position of the center of mass.

Example

For ease set origin to one of the particles

Example

ohh

ohh

i

iicm

ohh

ohh

i

iicm

i

iicm

mmm

mdmdm

m

myy

mmm

mdmdm

m

mxx

m

mr

mmm

mrmrmrr

)0()5.52sin()5.52sin(

)0()5.52cos()5.52cos(

...

...

321

332211

Example

0)0()5.52sin()5.52sin(

105.6)5.52cos(2 12

ohh

ohh

i

iicm

ohh

h

i

iicm

mmm

mdmdm

m

myy

mxmmm

dm

m

mxx

1) if there is an axis of symmetry, the center of mass will lie along the axis.

2) the center of mass can be outside of the body

Center of Mass

The point of an object which gravity can be thought to act.

This is conceptually different from center of mass

For now the center of gravity of an object is also it’s center of mass.

Center of Gravity

Motion of Center of Mass

Motion of Center of Mass

pmvmv

m

mv

mmm

mvmvmvv

dt

mrmrmrd

mmmdt

rd

mmm

mrmrmr

dt

d

dt

rd

m

mr

mmm

mrmrmrr

iiicm

i

iicm

cm

cm

i

iicm

...

...

...)(

...

1

...

...

...

...

321

332211

332211

321

321

332211

321

332211

External Forces and Center of Mass

cmiext

ernalexternal

iicmi

i

iicm

cm

cm

amF

FFF

Fmaam

m

ma

mmm

mamamaa

mmm

mvmvmv

dt

d

dt

vd

mmm

mvmvmvv

)(

)(

...

...

...

...

...

...

int

321

332211

321

332211

321

332211

Center of mass computations useful for when mass of a system changes with time

Rockets

Orbits

James and Ramon are standing 20.0 m apart on a frozen pond. Ramon has a mass of 60.0 kg and James has mass of 90.0 kg. Midway between the two is a mug of their favourite beverage. They pull on the ends of a light rope. When James has moved 6.0 m how far has Ramon moved?

Example

No external forces! Center of Mass will not move!

mr

m

mrr

cm

i

iicm

2150

300

6090

)60(10)90(10

Center of Mass will not move! James moved 6m to the right

mr

x

mx

m

mrr

cm

i

iicm

160

)90(4)6090(

26090

)60()90(4

A 1200 kg station wagon is moving along a straight highway at 12.0 m/s. Another car with mass 1800kg and speed 20.0 m/s has its center of mass 40.0 m away. (a) Find the position of the center of mass of the two cars. (b) Find magnitude of total momentum of the system. (c) Find the speed of the center of mass of the system. (d) Find total momentum using center of mass.

Problem – Young and Freedman 8.50

vectors again. equations for motion along one dimension

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