• Definition of termsDefinition of terms• Finding the Resultant by Graphical and Analytical Finding the Resultant by Graphical and Analytical

SolutionSolution

Vectors and ScalarsVectors and Scalars

Scalar Quantity – Scalar Quantity – a quantity that can be completely described by a quantity that can be completely described by giving its magnitude only. giving its magnitude only. Example: speed, time, energy, temperatureExample: speed, time, energy, temperature

Vector Quantity – Vector Quantity – a quantity that can be completely described by a quantity that can be completely described by giving its magnitude and direction. giving its magnitude and direction. Example: velocity, force, acceleration, momentumExample: velocity, force, acceleration, momentum

Resultant Vector – Resultant Vector – a single vector which has the same effect has a single vector which has the same effect has two or more vectors combined. two or more vectors combined.

Equilibrant Vector Equilibrant Vector – a single vector which balances two or more – a single vector which balances two or more vectors combined. Has the same magnitude as the resultant vector vectors combined. Has the same magnitude as the resultant vector but in the opposite direction.but in the opposite direction.

Direction ConventionDirection Convention

The following are positive The following are positive directions by convention: up, right, directions by convention: up, right, forward, north, and east.forward, north, and east.

The following are negative The following are negative directions by convention: down, left directions by convention: down, left backward, south, and west. backward, south, and west.

NN

EEWW

SS

NENE

ENEN

NWNW

WNWN

SESE

ESES

SWSW

WSWS

Methods of Vector Addition: Finding the ResultantMethods of Vector Addition: Finding the Resultant

A.A. Graphical Method Graphical Method A.1 Polygon Method A.1 Polygon Method

1.1. Draw (to scale) the vectors Draw (to scale) the vectors head-to-tail. head-to-tail.

2.2. The resultant is drawn from the The resultant is drawn from the beginning of the first vector to beginning of the first vector to the end of the last vector. the end of the last vector.

3.3. The magnitude and direction of The magnitude and direction of the resultant are measured the resultant are measured using a ruler and a protractor. using a ruler and a protractor.

yy

xx

AA

AA

BB

BB

RR

RR

A. Graphical Method A.2 Parallelogram Method - applicable to 2 vectors only

1.1. Draw (to scale) the vectors at Draw (to scale) the vectors at common origin (point o). common origin (point o).

2.2. Make a parallelogram by Make a parallelogram by constructing the sides parallel constructing the sides parallel to the 2 vectors.to the 2 vectors.

3.3. The resultant is drawn from The resultant is drawn from point o to meeting point of the point o to meeting point of the two parallel sides. two parallel sides.

4.4. The magnitude and direction of The magnitude and direction of the resultant are measured the resultant are measured using a ruler and a protractor. using a ruler and a protractor.

o

A

B

R

B. Analytical Method B. Analytical Method B.1 Component Method B.1 Component Method

1.1. Resolve each vector in Resolve each vector in their x and y components.their x and y components.

2.2. Get the summation of all Get the summation of all x – components x – components x and x and the summation of all y – the summation of all y – components components y. y.

3.3. Use the Pythagorean Use the Pythagorean Theorem to find the Theorem to find the resultant of the vectors.resultant of the vectors.

4.4. Use trigonometric Use trigonometric functions (sine, cosine, or functions (sine, cosine, or tangent) to find the angle tangent) to find the angle of the direction of the of the direction of the resultant. resultant.

yy

xx

AA

AA

BB

BB

RR

RR

AAyy

AAxx

BB

BByy

BBxx

xx

yy

A x x = A cosAA

A y y = A sinAA

AA

AAyy

AAxx

AA

B x x = B cosBB

B y y = B sinBB

BB

BB

BByy

BBxx

RR

RR

xx

yyx x = A = Axx + B+ Bxx

y y = A = Ayy + B+ Byy

R = xx22 + + yy

22

tan tan R R = y y / / xx

sin sin R R = y y / R / R

cos cos R R = x x / R / R

yy

xx

AA

BB

BB

RR

RR

AAyy

AAxx

BB

BByy

BBxx

xx

yy

AA

B. Analytical Method B. Analytical Method B.2 Triangle Method B.2 Triangle Method - applicable to 2 vectors only - applicable to 2 vectors only

1. Draw the vectors head-to-tail. Draw the vectors head-to-tail. (scale not necessary)(scale not necessary)

2.2.The resultant is drawn from the The resultant is drawn from the beginning of the first vector to the beginning of the first vector to the end of the last vector. end of the last vector.

3.3.The magnitude and direction of The magnitude and direction of the resultant are determined using the resultant are determined using cosine or sine law.cosine or sine law.

yy

xx

A

B

R

r

a

b

EXAMPLE: 1. A small boat travels 80.0 km north and then travels EXAMPLE: 1. A small boat travels 80.0 km north and then travels 60.0 km east in 1.0 hr. (a) What is the boat’s displacement for one-60.0 km east in 1.0 hr. (a) What is the boat’s displacement for one-hour trip? (b) What is the boat’s average speed during the one-hour hour trip? (b) What is the boat’s average speed during the one-hour trip?trip?

(a) x x = 60 km = 60 km + 0 = 60 km+ 0 = 60 km

R = R = ((60km)60km)22 + (80km) + (80km)22 = 100 km = 100 km

R R = tan -1 (80 km / 60 km) = 53 / 60 km) = 53oo

y y = 0 + 80 km = 80 km = 0 + 80 km = 80 km

RR = 100 km, 53= 100 km, 53oo NE NE

RR

RR

(b) Vave = (80 km + 60 km) / 1 hr

Vave = 140 km / hr

Finding the Resultant

Finding the Resultant