Higher Physics – Unit 1

1.1 Vectors

Scalars and VectorsScalars

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

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

Vectors

Here are some vector and scalar quantities:

Scalar Vector

time forcetemperature weight

volume accelerationdistance displacement

speed velocityenergymass

frequencypower

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

momentumimpulse

75 kmEdinburg

h

Distance and Displacement

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

300 km200 km

Inverness

Glasgow

N

E

S

W

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

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

distance - total distance travelled along a route

displacement - final position relative to starting position

km 500distance

)(270 West due km 75ntdisplaceme

DistanceDistance travelled by the helicopter:

DisplacementHelicopters final position relative to starting position:

Summary

Distance has only size, whereas displacement has both size and

direction.

Speed and VelocitySpeed is the rate of change of distance:

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

timedistancespeed

1h km 2502500speed

Velocity however, is the rate of change of displacement:

So for the 2 hour journey, the velocity is:

timentdisplacemevelocity

)(270 west due h km 37.5275velocity 1

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

Worksheet – Scalars and Vectors

Q1 – Q9

Vector AdditionVectors 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 1

Vector 2

RESULTANT VECTOR

Vectors can be added using a vector diagram.

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

Vector 1

Vector 2

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.

Resultant of a Vector

Example 1A 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.

N

E

S

W

40 m

50 m

Vectors are joined “ tip-to-

tail ”

(a) Draw a diagram showing the journey.

(b) Calculate the total distance travelled.

5040distance

m 90

(c) Calculate the total displacement of the person.

40 m

50 mdisplacemen

t

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

SizeBy Pythagoras:

222 cba

222 5040ntdisplaceme

41004100ntdisplaceme

m 64

Direction

adjoppθ tan

4050θ tan

θ

1.25tanθ 1

51.3θ

So the total displacement of the man is:

141.3 of bearing a on 64ms

90 + 51.3 = 141.3° (bearing)

40 m

50 mdisplacemen

t

1ms 1.5speed

timedistancespeed

6090speed

(d) Calculate the speed of the man.

(e) Calculate the velocity of the man.

timentdisplacemevelocity

6064velocity

141.3 of bearing a on ms1.07 velocity 1

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

Example 2A 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. N

E

S

W

20 ms-

1

5 ms-1

SizeBy Pythagoras 222 520v

222 cba

425425v

Direction

θ

adjoppθ tan

205θ tan

0.25tanθ 1

90 – 14 = 076° (bearing)

-1ms 20.6v14θ

076 of bearing on ms 20.6velocity -1

velocity

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°]

Worksheet – Vector Addition

Q1 – Q12

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 1An 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

E

S

W

Step 1Choose a suitable scale.

25 km : 1 cm

Step 2Draw diagram using a pencil and a protractor.

Step 3Measure the length of the resultant vector and convert using your scale.

Step 4Measure the size of the angle using a protractor.

4 cm

10 cm

13.7 cm

13.7 x 25 km = 342.5 km θ

Example 2A 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)tvd 0.510

km 5d

tvd 1.58km 12d

Stage 1 Stage 2 Total

km17 d

(b) N

E

S

W

1 km : 1 cm

5 cm

12 cm14.4 cm

θ

52 of θ angle with km 14.4ntdisplaceme

Length of Vector14.4 x 1 km = 14.4 km

Direction of Vectorθ = 52°

Answer Range14.5 km ± 0.4

km52° ± 2°

Worksheet – Vector Addition(Scale Diagram)

Q1 – Q3

Resolution of VectorsHorizontal and Vertical ComponentsTo 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=

VHVVVV

VH

OR

V

VH

VVθVVθ sin V

θ sinV VV VVθ cos H

θ cosV VH

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

Example 1A ship is sailing with a velocity of 50 ms-1 on a bearing of 320°.Calculate its component velocity(a) north

N

E

S

W

40°50 ms-

1

VN

VW

VVθ cos N

50V40 cos N

360° - 320° = 40°40 cos50VN

-1N ms 38.3V

(b) west

VVθ sin W 40°50 ms-

1

VN

VW

50V40 sin W

40 sin50VW 1

W ms 32.1V

Example 2A 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.

Horizontal16 ms-1

VH

VV30°

VVθ cos H

16V30 cos H

30 cos 16VH 1

H ms 13.9V

Vertical16 ms-1

VH

VV30°

VVθ sin V

16V30 sin V

30 sin16VV 1

V ms 8V

Slopes – Parallel and Perpendicular ComponentsOn a slope, the components of a vector are parallel and perpendicular to the slope.

g mW θ

resultantx

y

θ

Vectors are joined “ tip-to-

tail ”

hypadjθ cos

mgxθ cos

Perpendicular Component

Parallel Component

hypoppθ sin

mgyθ sin

θ cos g mx θ sin g my

Example 1A 10 kg mass sits on a 30° slope.Calculate the component of weight acting down (parallel) the slope.

hypoppθ sin

mg(resultant)

30 x

y

10 kg

g mW 30

mgy30 sin

9.810y30 sin

30 sin98y

N 49y

Worksheet – Resolution of Vectors

Q1 – Q8