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Higher Physics – Unit 1. 1.1 Vectors. A scalar quantity requires only size (magnitude) to completely describe it. A vector quantity requires size (magnitude) and a direction to completely describe it. Scalars and Vectors. Scalars. Vectors.

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Scalars and vectors

A scalar quantity requires only size (magnitude) to completely describe it.

A vector quantity requires size (magnitude) and a direction to completely describe it.

Scalars and Vectors

Scalars

Vectors


Here are some vector and scalar quantities:

time

force

temperature

weight

volume

acceleration

distance

displacement

speed

velocity

energy

momentum

mass

impulse

frequency

power

** Familiarise yourself with these scalar and vector quantities **


Distance and displacement

N

W

E

S

Distance and Displacement

A helicopter takes off from Edinburgh and drops a package over Inverness before landing at Glasgow as shown.

Inverness

To calculate how much fuel is needed for the journey, the total distance is required.

200 km

300 km

If the pilot wanted to know his final position relative to his starting position, the displacement is required.

75 km

Edinburgh

Glasgow


distance - total distance travelled along a route

displacement - final position relative to starting position

Distance

Distance travelled by the helicopter:

Displacement

Helicopters final position relative to starting position:


Distance has only size, whereas displacement has both size and direction.

Summary


Speed and velocity
Speed and Velocity

Speed is the rate of change of distance:

Say the helicopter journey lasted 2 hours, the speed would be:


Speed has only size, whereas velocity has both size and direction.

Velocity however, is the rate of change of displacement:

So for the 2 hour journey, the velocity is:



Vector addition
Vector Addition

Vectors are represented by a line with an arrow.

The length of the line represents the size of the vector.

The arrow represents the direction of the vector.

The sum of two or more vectors is called the resultant.


Vector 2

Vector 2

Vector diagrams are drawn so that vectors are joined “tip-to-tail”

Vector 1

Vector 1

RESULTANT VECTOR

RESULTANT VECTOR

The resultant of a number of forces is that single force which has the same effect, in both magnitude and direction, as the sum of the individual forces.

Vectors can be added using a vector diagram.

Resultant of a Vector


N

W

E

S

40 m

50 m

Vectors are joined “ tip-to-tail ”

Example 1

A man walks 40 m east then 50 m south in one minute.

(a) Draw a diagram showing the journey.

(b) Calculate the total distance travelled.

(c) Calculate the total displacement of the man.

(d) Calculate his average speed.

(e) Calculate his velocity.

(a) Draw a diagram showing the journey.


40 m

50 m

(b) Calculate the total distance travelled.

(c) Calculate the total displacement of the person.

Size

By Pythagoras:

displacement

The displacement is the size and direction of the line from start to finish.


displacement

40 m

50 m

90 + 51.3 = 141.3° (bearing)

Direction

So the total displacement of the man is:

(d) Calculate the speed of the man.


Speed has only size, whereas velocity has both size and direction.

(e) Calculate the velocity of the man.


velocity

N

W

E

5 ms-1

S

20 ms-1

90 – 14 = 076° (bearing)

Example 2

A plane is flying with a velocity of 20 ms-1 due east. A crosswind is blowing with a velocity of 5 ms-1 due north.

Calculate the resultant velocity of the plane.

Size

By Pythagoras

Direction


Q1. A person walks 65 m due south then 85 m due west.

(a) draw a diagram of the journey

(b) calculate the total distance travelled

(c) calculate the total displacement.

Q2. A person walks 80 m due north, then 20 m south.

(a) draw a diagram of the journey

(b) calculate the total distance travelled

(c) calculate the total displacement.

Q3. A yacht is sailing at 48 ms-1 due south while the wind is blowing at 36 ms-1 west.

Calculate the resultant velocity.

[ 150 m ]

[ 107 m at bearing of 232.6°]

[ 100 m ]

[ 60 m due north]

[ 60 ms-1 on bearing of 216.9°]



Vector addition scale diagrams
Vector Addition Scale Diagrams

Vectors are not always at right angles with each other.

To add such vectors together, it is easiest to use a scale diagram.

Example 1

An aircraft travels due north for 100 km. The aircraft changes its course to 25° west of north and travels for a further 250 km.

Find the displacement of the aircraft.


N

10 cm

W

E

S

4 cm

θ

Step 1

Choose a suitable scale.

25 km : 1 cm

Step 2

Draw diagram using a pencil and a protractor.

13.7 cm

Step 3

Measure the length of the resultant vector and convert using your scale.

13.7 x 25 km = 342.5 km

Step 4

Measure the size of the angle using a protractor.


Example 2

A ship sailing due west passes buoy X and continues to sail west for 30 minutes at a speed of 10 km h-1.

It changes its course to 20° west of north and continues on this course for 1½ hours at a speed of 8 km h-1 until it reaches buoy Y.

(a) Show that the ship sails a total distance of 17 km between marker buoys X and Y.

(b) By scale drawing or otherwise, find the displacement from marker buoy X to marker buoy Y.

(a)

Stage 1

Stage 2

Total


N

W

E

12 cm

S

θ

5 cm

(b)

1 km : 1 cm

Length of Vector

14.4 x 1 km = 14.4 km

14.4 cm

Direction of Vector

θ = 52°

Answer Range

14.5 km ± 0.4 km

52° ± 2°


Worksheet – Vector Addition

(Scale Diagram)

Q1 – Q3


Resolution of vectors
Resolution of Vectors

VV

VH

VV

VH

Horizontal and Vertical Components

To analyse a vector, it is essential to ‘break-up’ or resolve a vector into its rectangular components.

The rectangular components of a vector are the horizontal and vertical components.

V

=

OR


V

VV

VH

The horizontal and vertical component of the vector can be calculated as shown.


N

W

E

S

VW

VN

50 ms-1

360° - 320° = 40°

Example 1

A ship is sailing with a velocity of 50 ms-1 on a bearing of 320°.

Calculate its component velocity

(a) north

40°


VW

VN

50 ms-1

40°

(b) west


Example 2

A ball is kicked with a velocity of 16 ms-1 at an angle of 30° above the ground.

Calculate the horizontal and vertical components of the balls velocity.

Horizontal

16 ms-1

VV

30°

VH


16 ms-1

VV

30°

VH

Vertical


y

Vectors are joined “ tip-to-tail ”

Slopes – Parallel and Perpendicular Components

On a slope, the components of a vector are parallel and perpendicular to the slope.

θ

x

resultant

θ

Perpendicular Component

Parallel Component


x

mg

(resultant)

30

y

10 kg

Example 1

A 10 kg mass sits on a 30° slope.

Calculate the component of weight acting down (parallel) the slope.

30



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