Physics 201: Lecture 4, Pg 1 Lecture 4 l Today: Chapter 3 Introduce scalars and vectors Perform basic vector algebra (addition and subtraction) Interconvert

Physics 201: Lecture 4, Pg 1

Lecture 4

Today: Chapter 3

Introduce scalars and vectors

Perform basic vector algebra (addition and subtraction)

Interconvert between Cartesian & Polar coordinates


An “interesting” 1D motion problem: A race

Two cars start out at a speed of 9.8 m/s. One travels along the horizontal at constant velocity while the second follows a 9.8 m long incline angled below the horizontal down and then an identical incline up to the finish point. The acceleration of the car on the incline is g sin (because of gravity & little g)

At what angles , if any, is the race a tie?

9.8 m 9.8 m

FinishStart


A race According to our dynamical equations

x(t) = xi + vi t + ½ a t2

v(t) = vi + a t A trivial solution is = 0° and the race requires 2 seconds. Horizontal travel

Constant velocity so t = d / v = 2 x 9.8 m (cos ) / 9.8 m/s t1 = 2 cos seconds

Incline travel

9.8 m = 9.8 m/s ( t2 /2) + ½ g sin t2 /2)2

1 = ½ t2+ ½ sin (½ t2)2 sin t2)2 + 4 t2 8 =0

sinsin842

sin2sin32164

2

t


Solving analytically is a challenge

02cossincos2 2

A sixth order polynominal, (if z=1, 1-1+4-8+4 =0)

sin1 and cosLet 2 -zz

sin

sin842cos2

0484 :Solve 246 zzzz


Solving graphically is easier

sinsin842

cos2 21

ttSolve graphically

0.2 1.00.60.40.0 1.41.20.8Angle (radians)

time

(se

c.)

1.6

1.2

0.8

0.4

0

2.0

cos2

sin

sin842

=0.63 rad= 36°

If < 36° then the incline is fasterIf < 36 ° then the horizontal track is faster


A science project

You drop a bus off the Willis Tower (442 m above the side walk). It so happens that Superman flies by at the same instant you release the bus. Superman is flying down at 35 m/s.

How fast is the bus going when it catches up to Superman?


A “science” project

You drop a bus off the Willis Tower (442 m above the side walk). It so happens that Superman flies by at the same instant you release the car. Superman is flying down at 35 m/s.

How fast is the bus going when it catches up to Superman?

Draw a picturey

t

yi

0


A “science” project

Draw a picture Curves intersect at

two points

y

t

yi

0

ttg

y Superman2 v

2

Supermanv 2

tg

Supermanv2g

t

SupermanSupermanbus v2v2

v g

gtg


Coordinate Systems and Vectors

In 1 dimension, only 1 kind of system, Linear Coordinates (x) +/-

In 2 dimensions there are two commonly used systems, Cartesian Coordinates (x,y) Circular Coordinates (r,)

In 3 dimensions there are three commonly used systems,Cartesian Coordinates (x,y,z)Cylindrical Coordinates (r,,z)Spherical Coordinates (r,)


Scalars and Vectors

A scalar is an ordinary number. Has magnitude ( + or - ), but no direction May have units (e.g. kg) but can be just a number Represented by an ordinary character

Examples: mass (m, M) kilogramsdistance (d,s) metersspring constant (k)

Newtons/meter


Vectors act like…

Vectors have both magnitude and a direction Vectors: position, displacement, velocity, acceleration Magnitude of a vector A ≡ |A|

For vector addition or subtraction we can shift vector position at will (NO ROTATION)

Two vectors are equal if their directions, magnitudes & units match.

A

B C

A = C

A ≠B, B ≠ C


Vectors look like...

There are two common ways of indicating that something is a vector quantity:

Boldface notation: A

“Arrow” notation:

A or

A

A


Scalars and Vectors

A scalar can’t be added to a vector, even if they have the same units.

The product of a vector and a scalar is another vector in the same “direction” but with modified magnitude

A BA = -0.75 B


Exercise Vectors and Scalars

A. my velocity (3 m/s)B. my acceleration downhill (30 m/s2)C. my destination (the lab - 100,000 m east)D. my mass (150 kg)

Which of the following is cannot be a vector ?

While I conduct my daily run, several quantities describe my condition


Vectors and 2D vector addition

The sum of two vectors is another vector.

A = B + CB

C A

B

C


2D Vector subtraction

Vector subtraction can be defined in terms of addition.

B - C

B

-C

B

-C

B - C

= B + (-1)C

B+C Different direction and magnitude !


Unit Vectors

A Unit Vector points : a length 1 and no units Gives a direction. Unit vector uu points in the direction of UU

Often denoted with a “hat”: u = û U = |U| û

û

x

y

z

ii

jj

kk

Useful examples are the cartesian unit vectors [ i, j, k ] or

Point in the direction of the x, y and z axes.

R = rx i + ry j + rz k

or

R = x i + y j + z k

]ˆ,ˆ,ˆ[ zyx


Vector addition using components:

Consider, in 2D, C = A + B. (a) CC = (Ax ii + Ay jj ) + (Bx i i + By jj ) = (Ax + Bx )ii + (Ay + By )

(b) CC = (Cx ii + Cy jj )

Comparing components of (a) and (b):

Cx = Ax + Bx

Cy = Ay + By

|C| =[ (Cx)2+ (Cy)2 ]1/2

CC

BxAA

ByBB

Ax

Ay


ExampleVector Addition

A.A. {{3,-4,2}

B. {4,-2,5}

C. {5,-2,4}

D. None of the above

Vector A = {0,2,1} Vector B = {3,0,2} Vector C = {1,-4,2}

What is the resultant vector, D, from adding A+B+C?


Converting Coordinate Systems (Decomposing vectors)

In polar coordinates the vector R = (r,)

In Cartesian the vector R = (rx,ry) = (x,y)

We can convert between the two as follows:

• In 3D

tan-1 ( y / x )

22 yxr

y

x

(x,y)

rry

rx

j i

sin

cos

yxr

ryr

rxr

y

x

222 zyxr


Motion in 2 or 3 dimensionsPosition

Displacement

Velocity (avg.)

Acceleration (avg.)

ffii trtr , and ,

if rrr

t

r

avg.v

t v

a avg.

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Physics 201: Lecture 4, Pg 1 Lecture 4 l Today: Chapter 3 Introduce scalars and vectors Perform basic vector algebra (addition and subtraction) Interconvert