PHYS 20 LESSONS

Unit 2: 2-D Kinematics

Projectiles

Lesson 1: 2-D Vectors

Adding vectors (Tail-to-tip)

Reading Segment #1:

2-D Vectors

To prepare for this section, please read:

Unit 2: p.2

A. 2-D VECTORS

Recall, when we dealt with 1-D vectors, there were only

two possible directions.

We made one direction positive and the other direction negative.

Now, we will deal with 2-D vectors, where there are many

more than two directions. Expressing its direction is more

complicated.

A1. Direction of 2-D Vectors

- there are two methods to express the direction of 2-D vectors

that we use in Physics 20:

1. NEWS (North, East, West, South)

- acute angles measured with respect to the

nearest axis

2. RCS (rectangular coordinate system)

- angles measured in standard position (from

the positive x-axis)

1. Angles in NEWS N V

= 50W E

S

Method:

The angle is always positive and less than 90 (acute)

The angle is described relative to a nearby axis

i.e.

= 50 N of E

(North of the East Axis)

e.g. V N

60 W E

S

How would express this angle in NEWS?

e.g. V N

60 W E

S

= 60 of

In the NEWS method:

- you always show the acute angle on the diagram

- then, you will place two letters after the angle

(these are shown as spaces)

e.g. V N

60 W E

S

= 60 of W

The second letter is the axis that the angle "touches".

In this case, the angle is touching the West axis.

e.g. V N

60 W E

S

= 60 N of W

The first letter is the direction the angle moves from

the reference axis.

In this case, the angle is moving towards the North.

e.g. V N

60 W E

S

= 60 N of W

We interpret this as:

"60 degrees North from the West axis"

Ex. 1 Express the following angles in NEWS.

a) N b) N

W E W E

40V1 70

S S V2

a) N

W E

V1 70

S

1 = 70 of

a) N

W E

V1 70

S

1 = 70 of S

The second letter is the axis the angle touches.

In this case, it touches the South axis.

a) N

W E

V1 70

S

1 = 70 W of S

The first letter is the direction the angle moves from the

reference axis. In this case, it moves towards the West.

a) N

W E

V1 70

S

1 = 70 W of S

We interpret this as "70 degrees West from the South axis"

b) N

W E

40 V2

S

2 = 40 of

b) N

W E

40 V2

S

2 = 40 of E

The second letter is the axis the angle touches.

In this case, it touches the East axis.

b) N

W E

40 V2

S

2 = 40 S of E

The first letter is the direction the angle moves from the

reference axis. In this case, it moves towards the South.

b) N

W E

40 V2

S

2 = 40 S of E

We interpret this as "40 degrees South from the East axis"

2. Angles in RCS y

-x x

V -y

Method:

The angle is measured in standard position

- starts from the positive x-axis

Counterclockwise angles are positive

Clockwise angles are negative

e.g. V y

60 -x x

-y

How would express this angle in RCS?

e.g. V y

60 start

-x x

-y

RCS angles always start at the positive x-axis.

e.g. V y

60 -x x

-y

If the angle is counterclockwise, then it is positive.

So, = 180 - 60 = 120

e.g. V y

60 -x x

-y

But if the angle is clockwise, then it is negative.

So, = -90 + -90 + -60 = -240

Ex. 2 Express the following angles in RCS.

a) y b) y

-x x -x x

40V1 70

-y -y V2

a) y

Start

-x x

V1 70

-y

RCS angles start from the positive x-axis.

a) y

-x x

V1 70 1

-y

If the angle is clockwise, then it is negative.

So, 1 = -90 + -70

= -160

a) y

1

-x x

V1 70

-y

If the angle is counterclockwise, then it is positive.

So, 1 = 90 + 90 + 20

= 200

b) y

start

-x x

40 V2

-y

RCS angles always start at the positive x-axis.

b) y

-x x

40 V2

-y

Clockwise angles are negative.

So, 2 = -40

b) y

2

-x x

40 V2

-y

Counterclockwise angles are positive.

So, 2 = -90 + -90 + -90 + -50

= -320

Practice Problems

Try these problems in the Physics 20 Workbook:

Unit 1 p. 4 #1

Reading Segment #2:

Adding 2-D Vectors (Tail-to-Tip)

To prepare for this section, please read:

Unit 2: p.3

B. ADDING 2-D VECTORS

A key skill in Physics 20 and 30 is to add 2-D vectors.

There are two methods that we use:

1. Tail-to-Tip

- this is especially good when the vectors

are perpendicular (at 90)

2. Components

- a more tedious, labour-intensive method,

but it works for all cases (including 3-D)

B1. Adding 2-D Vectors (Tail -to-Tip)

Method:

Place the tail of the second vector on the tip of the first

i.e. "place one right after the other"

The resultant (or sum) vector R is drawn from the origin

to the tip of the second vector

i.e. Resultant is the "start to finish" vector

Solve the resulting triangle

If the vectors are at right angles, then you can use

Soh Cah Toa and the Pythagorean formula

Note:

Notice that the resultant vector is a "start-to-finish" vector.

This is the same description for the overall displacement

vector ( d ).

Thus, displacement is a good example of adding 2-D vectors.

Ex. 3 From a hunting lodge, a hiker walks the following path:

1.8 km South

then, 1.10 km East

Find the overall displacement of the hiker.

Include both magnitude and direction in your answer.

Strategy:

Overall displacement is a start-to-finish vector,

just like the resultant vector.

So, we will add the vectors tail-to-tip to get out answer.

1.8 km

1.1 km

Place the tail of the second vector onto the tip of the

first vector.

That is, place one vector right after the other.

1.8 km R

1.1 km

The resultant (displacement) is the "start-to-finish" vector.

Place the angle at the base (origin) of the resultant vector.

1.8 km R

901.1 km

Since it is a right triangle, we can use Soh Cah Toa and

the Pythagorean formula to find R and .

Pythag:

c2 = a2 + b2 1.8 km R c

a

1.1 km

b

Remember, c is always the hypotenuse (the longest side).

c2 = a2 + b2 1.8 km R c

a

R2 = (1.8 km)2 + (1.1 km)2

1.1 km

R2 = 4.45 km2 b

R = 4.45 km2

= 2.1 km

Soh Cah Toa:

1.8 km R hyp

adj

1.1 km

opp

Remember:

- the hypotenuse is the longest side

- the opposite side is the side furthest from (i.e. the one not touching the angle )

- the adjacent side is the side right beside the angle

Toa:

tan = opp 1.8 km R hyp

adj adj

1.1 km

opp

Since we know the opposite and the adjacent sides,

we will use tangent.

Toa:

tan = opp 1.8 km R hyp

adj adj

tan = 1.1 1.1 km

1.8 opp

= tan -1 (0.6111)

= 31

N

W E

31 R = 2.1 km

S

The answer in NEWS:

d = R = 2.1 km at 31 E of S

(or 2.1 km at 59 S of E)

y

-x x

31 R = 2.1 km

The answer in RCS: -y

d = R = 2.1 km at 301

(or 2.1 km at -59)

Animation:

2-D Addition (Tail-to-Tip)

1. http://www.phy.ntnu.edu.tw/java/vector/vector.html

2. http://www.walter-fendt.de/ph11e/resultant.htm

The second animation deals with forces,

but it shows vector addition very well.

Ex. 4 A boat takes the following course:

3.60 km West

then, 5.20 km North

Find the overall displacement of the boat.

Include both magnitude and direction in your answer.

5.20 km

3.60 km

Place the tail of the second vector onto the tip of the

first vector.

That is, place one vector right after the other.

5.20 km R

3.60 km

The resultant displacement vector is the "start-to-finish" vector.

Be certain to show the angle at the base (start) of the

resultant vector.

Pythag:

c2 = a2 + b2 5.20 km R

R2 = (3.60 km)2 + (5.20 km)2 3.60 km

R = 40 km2

= 6.32 km

Soh Cah Toa:

tan = opp 5.20 km R

adj

tan = 5.20 3.60 km

3.60

= tan -1 (1.4444)

= 55.3

N

6.32 km

55.3 W E

S

The answer in NEWS:

d = R = 6.32 km at 55.3 N of W

(or 6.32 km at 34.7 W of N)

y

6.32 km

55.3

-x x

-y

The answer in RCS:

d = R = 6.32 km at 125

(or 6.32 km at -235)

Ex. 5 A person walks the following path:

3.0 km at 20 S of E

then, 5.0 km at 10 W of S

Sketch the resultant displacement vector.

No calculations required.

20

3.0 km

Sketch the first vector:

3.0 km at 20 South from the East axis

20

3.0 km

Next, place new axes at the tip of the vector.

20

3.0 km

10

5.0 km

Add the second vector onto the tip of the first vector.

5.0 km at 10 West of the South axis

20

3.0 km

R

10

5.0 km

The resultant is the "start to finish" vector.

Practice Problems

Try these problems in the Physics 20 Workbook:

Unit 2 p. 4 #2 - 7