Physics is the Science of Physics is the Science of MeasurementMeasurement

We begin with the measurement of length: its magnitude and its direction.

We begin with the measurement of length: its magnitude and its direction.

LengtLengthh

WeighWeightt

TimeTime

Distance: A Scalar Distance: A Scalar QuantityQuantity

A scalar quantity:

Contains magnitude only and consists of a number and a unit.

A

B

DistanceDistance is the length of the actual is the length of the actual path taken by an object.path taken by an object.

DistanceDistance is the length of the actual is the length of the actual path taken by an object.path taken by an object.

distance = 20 m

Displacement—A Vector Displacement—A Vector QuantityQuantity

A vector quantity:

Contains magnitude AND direction, a number, unit & angle.

(12 m, 300)

A

BD = 12 m, 20o

• DisplacementDisplacement is the straight-line is the straight-line separation of two points in a separation of two points in a specified direction.specified direction.

• DisplacementDisplacement is the straight-line is the straight-line separation of two points in a separation of two points in a specified direction.specified direction.

Distance and Distance and DisplacementDisplacement

Net Net displacement:displacement:4 m,E4 m,E

6 6 m,Wm,W

D

What is the What is the distance traveled?distance traveled?

10 m !!

DD = 2 m, W= 2 m, W

• DisplacementDisplacement is the change of is the change of position based on the starting position based on the starting point. Consider a car that travels 4 point. Consider a car that travels 4 m, E then 6 m, W.m, E then 6 m, W.

• DisplacementDisplacement is the change of is the change of position based on the starting position based on the starting point. Consider a car that travels 4 point. Consider a car that travels 4 m, E then 6 m, W.m, E then 6 m, W.

xx = +4= +4xx = -2= -2

Identifying DirectionIdentifying Direction

A common way of identifying direction A common way of identifying direction is by reference to East, North, West, is by reference to East, North, West, and South. (Locate points below.)and South. (Locate points below.)

A common way of identifying direction A common way of identifying direction is by reference to East, North, West, is by reference to East, North, West, and South. (Locate points below.)and South. (Locate points below.)

40 m, 5040 m, 50oo N of E N of E

EW

S

N

40 m, 60o N of W40 m, 60o W of S40 m, 60o S of E

Length = 40 m

5050oo60o

60o60o

Identifying DirectionIdentifying Direction

Write the angles shown below by using Write the angles shown below by using references to east, south, west, north.references to east, south, west, north.Write the angles shown below by using Write the angles shown below by using references to east, south, west, north.references to east, south, west, north.

EW

S

N45o

EW

N

50o

S

Click to see the Answers . . .Click to see the Answers . . .500 S of E500 S of E

450 W of N450 W of N

Rectangular CoordinatesRectangular Coordinates

Right, up = (+,+)

Left, down = (-,-)

(x,y) = (?, ?)

x

y

(+3, (+3, +2)+2)

(-2, +3)(-2, +3)

(+4, -3)(+4, -3)(-1, -3)(-1, -3)

Reference is made Reference is made to to xx and and yy axes, axes, with with ++ and and -- numbers to numbers to indicate position in indicate position in space.space.

++++

----

Trigonometry ReviewTrigonometry Review

• Application of Trigonometry to Application of Trigonometry to Vectors Vectors

y

x

R

y = R sin y = R sin

x = R cos x = R cos

siny

R

cosx

R

tany

x R2 = x2 +

y2

R2 = x2 + y2

TrigonometryTrigonometry

Example 1:Example 1: Find the height of a Find the height of a building if it casts a shadow building if it casts a shadow 90 m90 m long and the indicated angle is long and the indicated angle is 3030oo..

90 m

300

The height h is opposite 300

and the known adjacent side is 90 m.

h

h = (90 m) tan 30o

h = 57.7 mh = 57.7 m

0tan 3090 m

opp h

adj

Finding Components of Finding Components of VectorsVectorsA component is the effect of a vector along other directions. The x and y components of the vector (R, are illustrated below.

x

yR

x = R cos

y = R sin

Example 2:Example 2: A person walks A person walks 400 m400 m in a direction of in a direction of 3030oo N of E N of E. How . How far is the displacement east and far is the displacement east and how far north?how far north?

x

yR

x = ?

y = ?400 m

E

N

The y-component (N) is OPP:

The x-component (E) is ADJ:

x = R cos y = R sin

E

N

Example 2 (Cont.):Example 2 (Cont.): A A 400-m400-m walk walk in a direction of in a direction of 3030oo N of E N of E. How . How far is the displacement east and far is the displacement east and how far north?how far north?

x = R cos

x = (400 m) cos 30o

= +346 m, E

x = ?

y = ?400 m

E

N Note:Note: xx is the side is the side adjacentadjacent to angle to angle

303000

ADJADJ = HYP x = HYP x CosCos 303000

The x-component The x-component is:is:RRxx = = +346 m+346 m

Example 2 (Cont.):Example 2 (Cont.): A A 400-m400-m walk walk in a direction of in a direction of 3030oo N of E N of E. How . How far is the displacement east and far is the displacement east and how far north?how far north?

y = R sin

y = (400 m) sin 30o

= + 200 m, N

x = ?

y = ?400 m

E

N

OPPOPP = HYP x = HYP x SinSin 303000

The y-component The y-component is:is:RRyy = = +200 m+200 m

Note:Note: yy is the side is the side oppositeopposite to angle to angle

303000

Example 2 (Cont.):Example 2 (Cont.): A A 400-m400-m walk walk in a direction of in a direction of 3030oo N of E N of E. How . How far is the displacement east and far is the displacement east and how far north?how far north?

Rx = +346 m

Ry = +200 m

400 m

E

NThe x- and y- The x- and y- components components are are eacheach + in + in

the first the first quadrantquadrant

Solution: The person is displaced 346 m east and 200 m north of the original

position.

Resultant of Perpendicular Resultant of Perpendicular VectorsVectorsFinding resultant of two perpendicular vectors is like changing from rectangular to polar coord.

R is always positive; is from + x axis

2 2R x y 2 2R x y

tany

x tan

y

x x

yR

Example 3:Example 3: A woman walks A woman walks 30 m, W30 m, W; ; then then 40 m, N40 m, N. Find her total . Find her total displacement.displacement.

= 59.1o N of W = 59.1o N of W

(R,) = (50 m, 126.9o)(R,) = (50 m, 126.9o)

040tan ; = 59.1

30

2 2( 30) (40)R R = 50 mR = 50 m

-30 m

+40 m R

Component MethodComponent Method

1. Start at origin. Draw each vector to 1. Start at origin. Draw each vector to scale with tip of 1st to tail of 2nd, tip scale with tip of 1st to tail of 2nd, tip of 2nd to tail 3rd, and so on for others.of 2nd to tail 3rd, and so on for others.

2. Draw resultant from origin to tip of last 2. Draw resultant from origin to tip of last vector, noting the quadrant of the vector, noting the quadrant of the resultant.resultant.

3. Write each vector in 3. Write each vector in x,yx,y components. components.

4. Add vectors algebraically to get 4. Add vectors algebraically to get resultant in resultant in x,yx,y components. Then components. Then convert to the total vector (convert to the total vector (R,R,).).

Example 4.Example 4. A boat moves A boat moves 2.0 km2.0 km east east then then 4.0 km4.0 km north, then north, then 3.0 km3.0 km west, west, and finally and finally 2.0 km2.0 km south. Find resultant south. Find resultant displacement.displacement.

EE

NN1. Start at origin. 1. Start at origin. Draw each vector Draw each vector to scale with tip of to scale with tip of 1st to tail of 2nd, 1st to tail of 2nd, tip of 2nd to tail tip of 2nd to tail 3rd, and so on for 3rd, and so on for others.others.2. Draw resultant from origin to tip of last 2. Draw resultant from origin to tip of last vector, noting the quadrant of the vector, noting the quadrant of the resultant.resultant.Note: The scale is approximate, but it is Note: The scale is approximate, but it is still clear that the resultant is in the fourth still clear that the resultant is in the fourth quadrant.quadrant.

2 km, 2 km, EE

AA

4 km, N4 km, NBB

3 km, W3 km, WCC2 km, 2 km,

SS

DD

Example 4 (Cont.)Example 4 (Cont.) Find resultant Find resultant displacement.displacement.3.3. Write each Write each

vector invector in i,ji,j notation:notation:A = +2 A = +2 xx

B = + 4 B = + 4 yyC = -3 C = -3 xx

D = - 2 D = - 2 yy 4.4. Add vectors A,B,C,D Add vectors A,B,C,D algebraically to get algebraically to get resultant inresultant in x,yx,y componentscomponents. .

RR ==

-1 -1 x x

+ 2 + 2 yy

1 km, west and 2 km north of origin..

1 km, west and 2 km north of origin..

EE

NN

2 km, 2 km, EE

AA

4 km, N4 km, NBB

3 km, W3 km, WCC2 km, 2 km,

SS

DD

5. 5. Convert to Convert to resultant resultant vector vector See next See next page.page.

Example 4 (Cont.)Example 4 (Cont.) Find resultant Find resultant displacement.displacement.

EE

NN

2 km, 2 km, EE

AA

4 km, N4 km, NBB

3 km, W3 km, WCC2 km, 2 km,

SS

DDResultant Sum Resultant Sum is:is:RR = -1 = -1 xx + 2 + 2 yy

Ry= +2 km

Rx = -1 km

RR

Now, We Find Now, We Find R, R, 2 2( 1) (2) 5R

R = 2.24 km

2 kmtan

1 km

= 63.40 N of W

Conclusion of Chapter 3B - Conclusion of Chapter 3B - VectorsVectors