Chapter 3 – Two Dimensional Motion and Vectors

3 – 1: Objectives

• Distinguish between a scalar and a vector

• Add and subtract vectors using the graphical method

• Multiply and Divide Vectors by Scalars

Every physical quantity is either a scalar or a vector quantity

• Scalar: a physical quantity that can be completely specified by its magnitude (a number) with appropriate units.– Examples: mass, speed, distance and volume

• Vector: a physical quantity that has both magnitude and direction– Examples: position, displacement, velocity,

and acceleration

Notation used to represent vector quantities

• Book uses boldface type to represent vector quantities– v– a– x– ∆x

• Handwritten – place a “vector symbol” over the variable– v– a– x– ∆x

Vectors can be represented by diagrams

• Arrows are used to show a vector quantity that points in the direction of the vector.

• The length of the arrow represents the magnitude

50 m/s, East

25 m/s, East

Notice, the 50 m/s vector is twice as long as the 25 m/s vector

Draw 2 vectors that represent 10 m east and 15 m west

Notice: The arrow head is pointing in the required direction and the lengths are drawn to a chosen scale where each unit represents 5 m.

Vector Addition – Graphical Method

• 1. Vectors to be added are physically placed tip – to – tail (the tip of one vector touches the tail of the next vector) in any order– NOTE: Within a diagram, vectors can be

moved (translated) for the purpose of vector addition, as long as the direction and the length remain the same.

Resultant Vector

• the sum of 2 or more vectors

• the solution to a vector addition problem

• also called vector sum

Finding a Resulant Vector

• Found graphically by drawing another vector that begins at the tail of the first vector and ends at the tip of the last vector that is being added.--NOT TIP-TO-TAIL! Beginning to end.

Graphical Vector Addition in One - Dimension

Tip – to - tail

NOTES: technically if all vectors are in one – dimension, they would be drawn on top of each other, these are separated slightly for clarity.

The magnitude of the resultant vector can be found by measuring the length and converting the number to the proper units using the given scale.

The direction is shown by the arrow tip.

Tail – to - tailTip – to - tip

• The diagram shown on the previous page shows 2 displacement vectors that were being added (10 m east and 15 m west)

• The resultant vector is obviously 5 m west.• In one – dimension it is certainly easier to

use the magnitude and a +/- sign for direction to add the vectors– Ex. (+10 m) + (-15 m) = -5 m

• The resultant vector is 5 m to the west!

Graphically Add the following 3 displacement vectors (1-dimensional)

• Choose an appropriate scale and draw the graphical solution to this vector addition problem

225 m north, 175 m south, and 125 m south

Graphical Vector Addition in 2 Dimensions

• The graphical procedure is the same as in 1 – dimension– Vectors to be added are physically placed tip – to –

tail (the tip of one vector touches the tail of the next vector) in any order

– The resultant vector is found graphically by drawing another vector that begins at the tail of the first vector and ends at the tip of the last vector

Vector 1

Plus vector 2Resultant vector

• The magnitude of the resultant vector can be found by measuring the length and converting the number to the proper units using the given scale. (exactly the same as in 1 – dimension)

• The direction is described differently.– The direction of a 2 – dimensional vector is

graphically determined with a protractor and is measured counter-clockwise (CCW) from the +x - axis

Vector 1

Plus vector 2Resultant vector

θ

1. Place an x/y coordinate system at THE TAIL of your resultant vector

2. Measure the direction CCW from the + x - axis

Important comment!

• If given a vector diagram where the vectors are not drawn tip - to – tail, you can move a vector in a diagram so that you can set up a tip – to – tail situation! Proceed as before.

Vector 1

Vector 2

Vector 1

Vector 2

Resultant Vector

Hints about vector addition

• When adding vectors:– 1. The vectors must represent the same

physical quantity (you can’t add velocity and displacement)

– 2. The vector quantities must have the same units (you can’t add m and km, you must convert first)

Resultant Vector

• The resultant vector represents a SINGLE vector that produces the same RESULT as the other vectors (addends) acting together

Example (Displacement)

3 m

4 m5 m

53º

Walking 3 m east and then 4 m north puts you at the same final position as walking 5 m at an angle of 53º

Sample problem

140 m

120 m

Find the resultant displacement.

• A person rows due east across the Delaware River at 8.0 m/s. The current carries the boat downstream (south) at 2.5 m/s. What is the person’s resultant velocity?

• Graphical Vector Addition Practice– Worksheet– Rulers– Protractors

Review Problems

1. Two ropes are tied to a tree to be cut down. The first rope pulls on the tree with a force of 350 N west. The second pulls at 425 N at 320 degrees. What’s the resultant force?

2. A person drives through town 6 blocks north, then 3 blocks east. They run into a one way street and have to travel 1 block south to go 2 more blocks east. Finally, the person parks and walks 2 blocks north to the destination. What is the person’s displacement?

Part II

Properties of Vectors

• 1. Vectors may be translated in a diagram (moved parallel to themselves)

• 2. Vectors may be added in any order (Vector addition is commutative)

• 3. To subtract a vector, add its opposite.– The opposite of a vector has the same magnitude and

points in the opposite direction. (+/- 180º)

• 4.Multiplying or dividing vectors by scalars results in vectors

2. Vectors may be added in any order (Vector addition is

commutative)

3. To subtract a vector, add its opposite.

A - B

A

B

A + (-B)

A

-B

4.Muliplying or dividing vectors by scalars results in vectors

A 2A A/2

Notice: The magnitude is multiplied or divided but the direction remains the same.

A ball is thrown 25 m at an angle of 30º

Two times this displacement vector is 50 m at an angle to 30º

Sample problems

• Given the following vectors:

A = 50 m South B = 80 m East

C = 65 m @ 210° D = 110 m @ 140°

Find:

1.A – C 2. 3D + B -2A

3. ½ B – 4A

3-2 Vector Operations

• Objectives:– Identify appropriate coordinate systems for

solving problems with vectors.– Apply the Pythagorean Theorem and tangent

function to calculate the magnitude and direction of a resultant vector.

– Resolve vectors into components using the sine and cosine functions.

– Add vectors that are not perpendicular

Geometry / Trigonometry Review

• Pythagorean Theorem – The square of the hypotenuse of a right triangle is equal to the sum of the squares of its legs

• c2 = a2 + b2

a

b

c

• Trigonometric Ratios:

Hypotenuse (c)

Leg (b)

Leg (a)

CA

B

Opposite side

hypotenuse

Adjacent side

hypotenuse

Opposite side

Adjacent sideSin θ =

Cos θ = Tan θ =

Using Trig. Ratios

• Given an acute angle of a right triangle, to find the ratio of 2 specific sides of the triangle, enter the appropriate function (sine, cosine, tangent) of the angle in your calculator.

a

b

c

20º

Sin(20º)=b/c

Cos(20º) = a/c

Tan(20º) = b/a

• To find an acute angle of a right triangle, enter the inverse of the appropriate function of the ratio of the 2 corresponding sides.

a

b

cθ

θ = sin-1(a/c)

θ = cos-1(b/c)

θ = tan-1(a/b)

• Trigonometry Review Practice Worksheet

Part III

Vector Addition – Analytical Method

• Case #1 (easiest method): – Adding 2 Vectors that are perpendicular

R = magnitude of the resultant vector

R = A2 + B2

Resultant, R

A

B

θ

The angle, θ, of the triangle can be found using the tan-1 function and THEN CONVERT it to the direction measured CCW from the +x - axis

Example for Case #1• Add the following 2 velocity vectors.

5 m/s west (180º) and 8 m/s north (90º)

5 m/s

8 m/sR

θ

R2 = 52 + 82

R = 9.4 m/s

θ = tan-1 (8/5)

θ = 58º

The direction (measured CCW from the +x – axis) is found by subtracting 180 – 58 = 122º

R = 9.4 m/s <122º

Case #2: Adding more than 2 perpendicular vectors

• First, find the vector sum of all of the horizontal vectors, call this Rx.

• Second, find the vector sum of all of the vertical vectors, call this Ry.

• Find the vector sum of Rx and Ry

– By following the method from Case #1

Example of Case #2

• A boyscout walks 8 m east, 2 m north, 6 m east, 10 m south, 3 m east, 5 m south and 3 m west.

Horizontal Vectors

+ 8m

+ 6m

+ 3m

- 3m

+14 m

Vertical Vectors

+ 2m

- 10m

- 5m

-13 m

R2 = 142 + 132

R = 19.1 mθ = tan-1 (13/14)θ = 43ºThe direction (measured CCW from the +x – axis) is found by subtracting 360 – 43 = 317º

Rx

Ry

R

θ

R = 19.1 m <317º

Vector Resolution (opposite process of adding 2 vectors)

• Any vector acting at an angle can be replaced with 2 vectors that act perpendicular to each other, one horizontal and one vertical. (The 2 vectors working together are equivalent to the single vector acting at an angle.)

Step 1

• Sketch the given vector with the tail located at the origin of an x-y coordinate system. (Ex. 25 m at an angle of 36º)

25 m

36º

Step 2

• Draw a line segment from the tip of the vector perpendicular to the x-axis

25 m

Notice, you now have a right triangle with a known hypotenuse and known angle measurements

36º

Step 3

• Replace the perpendicular sides of the right triangle with vectors drawn tip – to - tail

25 m

Step 4

• Use sine and cosine functions to find the horizontal and vertical components of the given vector.

25 m

Rx

Ry36º

Cos(36) = Rx/25

Rx = 25cos(36)

Rx = 20.2 m

sin(36) = Ry/25

Ry = 25sin(36)Ry = 14.7 m

Important

• Remember that the 2 components acting together gives the same result as the single vector acting at an angle.

• ****The 2 components can be used to REPLACE the single vector****

Example #2

• Find the components of 16m at 200º

200º

20º

You have 2 choices at this point.

You can use the directional angle of 200 and not worry about the sign of the components (the calculator will do it for you).

OR, you can use 20 and YOU must remember to put – signs when the component points down or to the left

Example #2

• Find the components of 16m at 200º

200º

20º

Rx = 16cos(200) = -15 m

Ry = 16sin(200) = -5.5 m

Rx = -16cos(20) = -15 m

Ry = -16sin(20) = -5.5 m

Case #3 – Adding Vectors at Angles (not perpendicular)

• When vectors to be added are not perpendicular, they do not form sides of a right triangle.

Look at the geometry for the situation

Rx1

Ry1

Rx2

Ry2

Rx1 + Rx2

Ry1 + Ry2

Resultant Vector

• Notice, the length of the horizontal component of the resultant vector is equal to the sum of the lengths of the horizontal components of the vectors that are being added together.

• This is also true for the vertical component.

Steps for solving Case #3 Problems

• 1. Resolve each vector that is being added (addends) into components.

• 2. Add all the horizontal components together and all the vertical components together (Case #2)

• 3. Use the Pythagorean Theorem and trig ratios to find the resultant vector (Case #1)

Example for Case #3

• Add these 2 vectors together: 10 m/s at 0º and 12 m/s at 25º (Find the resultant vector, R at θ)

10 m/s

12 m/s

25º

R

θ

Example for Case #3

Find components of each vector

10 m/s

12 m/s

25º

R

θ

Vector 1

x y

10cos(0) 10sin(0)

Vector 2 12cos(25) 12sin(25)

Example for Case #3

Add horizontal and vertical components

10 m/s

12 m/s

25º

R

θ

Vector 1

x y

10cos(0) 10sin(0)

Vector 2 12cos(25) 12sin(25)

21 5.1

Example for Case #3

Find the magnitude of the resultant vector using the Pythagorean Theorem

R

θ

R2 = 212 + 5.12

R = 21.6 m/s

θ = tan-1 (5.1/21)

θ = 14º

R = 21. m/s at 14º21

5.1