Ph i 218Physics 218yChapter 3&4Chapter 3&4

Prof. Rupak pMahapatra

Physics 218, Chapter 3 and 4 1

Checklist for Todayy•Clickers can be registered on elearning

Will it till W d b f i–Will wait till Wed before using

D EOC Ex i f m Ch p 3 4 thi •Do EOC Exercises from Chap 3, 4 this before Monday

•Quiz on Chapter 2 in Recitation

•Google group set up for class discussions/help

Physics 218, Chapter 3 and 4 2

VectorsVectors•Why we care •Why we care

b t thabout them•Addition & •Addition & S bt tiSubtraction

Physics 218, Chapter 3 and 4 3

Why do we care about Vectors?yAs you may have noticed, the world is not one-dimensionalone-dimensional

• Three dimensions: X, Y and Z. • Example: • Example:

1.Up from us2 Straight in front of us2.Straight in front of us3.To the side from us

– All at 90 degrees from each other All at 90 degrees from each other. Three dimensional axis.

• Need a way of saying how much in each d

y y gdirectionFor this we use VECTORS

Physics 218, Chapter 3 and 4 4

For this we use VECTORS

Vector and Scalar

• Vectors have a magnitude Vectors have a magnitude AND a direction–I’m driving 70 miles/hr S thE t t H tSouthEast to Houston

• Scalars are just a number• Scalars are just a number–My speedometer says 70 m/hrMy speedometer says 70 m/hr

Physics 218, Chapter 3 and 4 5

Where am I?Let’s say I’m here You’re here (origin)I call you on the cell

phone. How do I tell you how to get to

My single vector in some

me?2 equivalent ways:

My single vector in somefunny direction, can be

thought of as two vectors 1)Travel 11.2 km at an

angle of 26.5 thought of as two vectors in nice simple directions(like X and Y). This can

gdegrees

2)Travel 10 km East Physics 218, Chapter 3 and 4 6

( )make things much easier

)then 5 km North

Re-write my locationy•Describe my location i f h in terms of the sum of two vectors

Θ|V|||VV V V YX

Θsin|V|||VΘcos|V|||VX

•Careful when using

Θsin|V|||VY

gthe sin and cos

Physics 218, Chapter 3 and 4 7

Specifying a Vectorp y g•Two equivalent ways:

Components V and V–Components Vx and Vy–Magnitude V and angle angle

•Switch back and forth–Magnitude of VMagnitude of V

|V| = (vx2 + vy

2)½Pythagorean Pythagorean

Theorem–Tan = vy /vxy x

Either method is fine, but you should pick which is easiest and

Physics 218, Chapter 3 and 4 8which is easiest, and be able to use both

Unit VectorsFormal notation for vectorsf

direction x the in 1 means i

direction y the in 1 means j

direction x the in 1 means i

jV iV V

direction y the in 1 means j

jV iV V yx

Physics 218, Chapter 3 and 4 9

Unit VectorsFormal notation also uses:

i as same the is x

j as same the is y yV xV V

j as same the is y

yV xV V yx

Physics 218, Chapter 3 and 4 10

Vector in Unit Vector Notation

|V|||VΘcos|V|||VX

V V VΘsin|V|||VY

j V i V V

V V V YX

jΘsin |V| iΘcos |V| V

j V i V V YX

j| || |

Physics 218, Chapter 3 and 4 11

General Addition ExamplepAdd two vectors Add two vectors using the i-h t d jhats and j-hatshats

DD D 21R

j k 5 i k 0 D

j km 0 i km 10 D1

j km 5 i km 10 D

j km 5 i km 0 D

R

2

Physics 218, Chapter 3 and 4 12

j m 5 m DR

Ch4: Motion in 2-Dimensions• Moving from Chapter 3 to g pChapter 4

• This is what all the setup has been for!has been for!

• Motion in two and three Motion in two and three dimensions– For now we’ll ignore air friction

Physics 218, Chapter 3 and 4 13

Position in 2 dimensions

R W

position, our as R Writeorigin the to relative

dimensions 2 in

jY iX R

jY iX R Physics 218, Chapter 3 and 4 14

Velocity in 2 dimensionsy then position, the is R If

dtRd V

p

dt)jY id(X

jdtdY idt

dX dt

jyV ixV dtdt

dtjd 0 dt

id used have We :Note

Physics 218, Chapter 3 and 4 15

dtdt

Acceleration in 2 dimensions then position, the is R If

dtVd a

)jV id(Vdt

dt

)jV id(V Yx

jdtVd idt

dV Yx

ja ixa dtdt

Physics 218, Chapter 3 and 4 16

jyx

Projectile MotionjThe physics of the universe:h phys cs of th un rsThe horizontal and The horizontal and The horizontal and The horizontal and

vertical Equations of vertical Equations of vertical Equations of vertical Equations of Motion behave Motion behave M nM nindependentlyindependentlyp yp y

This is why we use vectors in the first placePhysics 218, Chapter 3 and 4 17

first place

How to use the Tricks and Methods

•Next we’ll do an example problem like one f th h k bl i th t t b kof the homework problems in the text book

•Solve this problem using the right method–Draw a diagram–Convert the numbers to variables–Solve to get a formula–Plug in the numbers at the endPlug in the numbers at the end–Check

•Reasonable numbers? •Reasonable numbers? •Silly numbers?A th t d th bl ?

Physics 218, Chapter 3 and 4 18•Another way to do the same problem?

How to Solve Problems

The trick for all The trick for all these problems is to these problems is to b k th i t break them up into

h X d Y the X and Y ddirections

Physics 218, Chapter 3 and 4 19

Ball Droppingpp gAnalyze Vertical yand Horizontal separately!!!separately!!!

• Ay = g(d d )(downwards)

• Ax = 0x– Constant for Both cases!!!cases!!!

Vx = 0 Vx>0Physics 218, Chapter 3 and 4 20

Vx 0 Vx 0

A weird consequenceqProve that an object projected projected horizontally will reach the will reach the ground at the

ti same time as an object jdropped vertically

Physics 218, Chapter 3 and 4 21

vert cally

Physics 218, Chapter 3 and 4 22

Checklist for Todayy•Do EOC Exercises from Chap 3, 4 this week

•Quiz

Physics 218, Chapter 3 and 4 23

Firing up in the air at an angleg p gA ball is fired up in the air with speed Vo and angle Ignore air friction The acceleration angle o. Ignore air friction. The acceleration due to gravity is g pointing down.What is the final velocity here?What is the final velocity here?

Physics 218, Chapter 3 and 4 24

Throw a brick out of a windowJen pushes a brick directly out of a directly out of a window that is a height h above the

d T A i g

ground. Two Aggies observe that brick hits the ground a hits the ground a distance D from the base of the building. A i b i k

hAssuming brick was moving horizontally at the top (and ignoring D

y=0the top (and ignoring air friction), how fast must Jen have pushed the brick out?

D

Physics 218, Chapter 3 and 4 25

pushed the brick out?

Kick a footballA football is kicked at angle with respect

t th d ith d V C l l tto the ground with speed V0. Calculate:a) The maximum heightb) Th l it t th i h i htb) The velocity at the maximum heightc) The time of travel before the football hits

the groundthe groundd) How far away it hits the ground (Range R)e) What angle maximizes the distance travelede) What angle maximizes the distance traveled

Assume the ball leaves the foot at ground level and ignore air resistancelevel and ignore air resistance

Physics 218, Chapter 3 and 4 26

Maximize Range Againg g• Find the minimum initial speed of a p

champagne cork that travels a horizontal distance of 11 metershorizontal distance of 11 meters.

Physics 218, Chapter 3 and 4 27

Rescue PlaneYou are the pilot of a rescue plane and you

fl i l ith h i t l d are flying a plane with a horizontal speed of Vp. Your mission is to drop supplies for some mountain climbers on a ridge a some mountain climbers on a ridge a vertical distance h below.

To do this you have to release the supplies a To do this you have to release the supplies a horizontal distance R in advance of the mountain climbersmountain climbers.

What vertical velocity should you give the supplies so that they arrive precisely at supplies so that they arrive precisely at the the climbers position?

Physics 218, Chapter 3 and 4 28

Rescue PlaneYou are the pilot of a rescue plane. p p

Your mission is to drop supplies to isolated mountain climbers on to isolated mountain climbers on a rocky ridge a height h below. If your plane is traveling If your plane is traveling horizontally with a speed of VO:

) f f h a) How far in advance of the recipients (horizontal distance) p ( )must the goods be dropped?

Physics 218, Chapter 3 and 4 29

Rescue Plane cont.b) Suppose instead, that your plane ) pp y p

can’t get to that position. Instead you must release the supplies a you must release the supplies a horizontal distance D in advance of the mountain climbers What verticalthe mountain climbers. What verticalvelocity should you give the supplies

th t th i i l t th so that they arrive precisely at the the climbers position.

c) What is the speed of the supplies as they hit the ground?

Physics 218, Chapter 3 and 4 30

they hit the ground?

Quiz! More Realistic KickQA football is kicked at angle 0 with a

d V Th b ll l th t speed V0. The ball leaves the punters foot h meters above the ground. g

How far does it travel, in the X direction before it hits the ground?direction, before it hits the ground?

hPhysics 218, Chapter 3 and 4 31

Next Week•Chapter 5: Read for Next timep

–Newton’s Laws of MotionHW2 d M d–HW2 due Monday

•Cover CH 3&4 in recitation •Cover CH. 3&4 in recitation, due the following Monday

Physics 218, Chapter 3 and 4 32

Physics 218, Chapter 3 and 4 33