1. VECTORS & RELATIVE
MOTION
1. Show vector direction as a bearing, compass direction or angle relative to a fixed
axis.
2. Perform simple vector arithmetic (including addition, subtraction)
3. Resolve vectors into components
4. Define the term relative velocity
5. Use vectors to solve relative velocity problems
Reading: Chapter 8 (p91 to 103)

2. VECTOR DIRECTION
Definition
A vector is a physical quantity that has both size and direction.
Examples
Displacement, velocity, acceleration and force.
A vector has a head and a tail head
head
tail tail
A vector’s direction can be described in a number of different ways:
(a)Bearings
A bearing is an angle measure clockwise from the North.
Eg.
N N
N
30o
45o
30o
Direction = 045o
Direction = Direction =

3. (b)Compass reference
An angle is given from the N, S, E or W direction
Eg.
N N
N
30o
45o
30o
Direction = Direction = Direction =
(c) Referenced from the vertical or horizontal
An angle is given from a vertical or horizontal axis
Eg.
20o
60o
Direction = Direction =

4. VECTOR ARITHMETIC
Addition
Two vectors are added head to tail to produce a resultant vector. The resultant vector
is a single vector that has the same effect as the two vectors combined.
Eg:
Adding two vectors, a and b:
head
head
b
~
a
~
tail
tail
a b
~+ ~
Ex.8A Q.1 to 8

5. Subtraction
Vector subtraction is “addition of the opposite”. The opposite of a vector is a vector
which has the same magnitude but is opposite in direction.
Eg:
Subtracting two vectors, a minus b:
b
~
a
~
-b
~
a b a - b
~ - ~ = ~ + ~

6. VECTOR COMPONENTS
Any vector can be drawn as the sum of two other vectors which are drawn at right
angles to each other. These two vectors are called components.
Examples - Resolving vectors into components:
1 Any vector can be expressed as the sum of two components
(a) (b) (c)
2 Horizontal and vertical components are the most useful components
30 N 30 N
Vertical
component 25o 25o
Horizontal
component The vector is the sum of its
horizontal and vertical components

7. CALCULATING THE SIZE OF THE COMPONENTS
3 It is possible to calculate the component of a vector along any axis
30 N
F
~
25o f
~
This is the component of the vector, F along this axis
~
f = F.cos25o
Example
Calculate the size of the component of the following velocity vector along the axis
shown:
50 ms-1
Exercises: “Vector arithmetic &
components”

8. CHANGE IN VELOCITY
The change in velocity of a moving object is the final velocity minus the initial
velocity:
∆v = vf - vi
Example
1. A tennis ball falls to the ground, striking at right angles. It bounces off the ground
and travels along the same path on the rebound as it travelled as it was falling.
The initial velocity is 10 ms-1 vertically downwards. It rebounds with a final velocity
of 10 ms-1 upwards. Calculate the change in velocity of the ball.
10 ms-1
10 ms-1

9. Example
2. A billiard ball strikes the cushion of a billiard table at an angle of 20o and rebounds
at the same angle. Use a vector diagram to calculate the change in velocity.
8 2 ms-1
20o
20o
2 ms-1
Vector subtraction Calculation
Ex.8A Q.9 to 14

10. Relative velocity is the velocity of an object in relation to another object.
This other object can be stationery (like the ground) or moving.
RELATIVE VELOCITY along a straight line
Consider the example of a train travelling in a straight line along a track.
A boy is standing still on the roof of a train which is travelling slowly in a straight line
at a speed of 2 kmh-1.
2 kmh-1
A second boy standing on the ground holding a speed gun measures the boy’s
velocity at 2 kmh-1.

11. Relative velocity is the velocity of an object in relation to another object.
This other object can be stationery (like the ground) or moving.
RELATIVE VELOCITY along a straight line
Consider the example of a train travelling in a straight line along a track.
A boy is standing still on the roof of a train which is travelling slowly in a straight line
at a speed of 2 kmh-1.
2k
A second boy standing on the ground holding a speed gun measures the boy’s
velocity at 2 kmh-1.

12. Relative velocity is the velocity of an object in relation to another object.
This other object can be stationery (like the ground) or moving.
RELATIVE VELOCITY along a straight line
Consider the example of a train travelling in a straight line along a track.
A boy is standing still on the roof of a train which is travelling slowly in a straight line
at a speed of 2 kmh-1.
2k
A second boy standing on the ground holding a speed gun measures the boy’s
velocity at 2 kmh-1.
The boy’s velocity is 2 kmh-1 relative to the ground.

13. A boy is standing still on the roof of a train which is travelling slowly in a straight
line at a speed of 2 kmh-1.
2 kmh-1
A second boy standing on the roof of the train measures the boy’s velocity at 0 kmh-1.

14. A boy is standing still on the roof of a train which is travelling slowly in a straight
line at a speed of 2 kmh-1.
2
A second boy standing on the roof of the train measures the boy’s velocity at 0 kmh-1.

15. A boy is standing still on the roof of a train which is travelling slowly in a straight
line at a speed of 2 kmh-1.
2
A second boy standing on the roof of the train measures the boy’s velocity at 0 kmh-1.
The boy’s velocity is 0 kmh-1 relative to the train.

16. TWO VELOCITIES RELATIVE TO EACH OTHER
Consider the example of a car (travelling at 20 kmh-1) overtaking a bus which is
travelling considerably slower (at 12 kmh-1).
Car 20 kmh-1
Bus 12 kmh-1
• The velocity of the car relative to the bus is 8 kmh-1 in the forward
direction because an observer in the bus sees the car moving forward (past the
bus) at 8 kmh-1.
• The velocity of the car relative to the bus is the velocity of the car minus the
velocity of the bus. We can write this as a simple equation:
vcb = vc - vb Where c = car
~ ~ ~ b = bus
The vector subtraction is shown below:

17. TWO VELOCITIES RELATIVE TO EACH OTHER
Consider the example of a car (travelling at 20 kmh-1) overtaking a bus which is
travelling considerably slower (at 12 kmh-1).
Car 20 kmh-1
Bus 12 kmh-1
• The velocity of the car relative to the bus is 8 kmh-1 in the forward
direction because an observer in the bus sees the car moving forward (past the
bus) at 8 kmh-1.
• The velocity of the car relative to the bus is the velocity of the car minus the
velocity of the bus. We can write this as a simple equation:
vcb = vc - vb Where c = car
~ ~ ~ b = bus
The vector subtraction is shown below:
20 kmh-1

18. TWO VELOCITIES RELATIVE TO EACH OTHER
Consider the example of a car (travelling at 20 kmh-1) overtaking a bus which is
travelling considerably slower (at 12 kmh-1).
Car 20 kmh-1
Bus 12 kmh-1
• The velocity of the car relative to the bus is 8 kmh-1 in the forward
direction because an observer in the bus sees the car moving forward (past the
bus) at 8 kmh-1.
• The velocity of the car relative to the bus is the velocity of the car minus the
velocity of the bus. We can write this as a simple equation:
vcb = vc - vb Where c = car
~ ~ ~ b = bus
The vector subtraction is shown below:
12 kmh-1
20 kmh-1

19. TWO VELOCITIES RELATIVE TO EACH OTHER
Consider the example of a car (travelling at 20 kmh-1) overtaking a bus which is
travelling considerably slower (at 12 kmh-1).
Car 20 kmh-1
Bus 12 kmh-1
• The velocity of the car relative to the bus is 8 kmh-1 in the forward
direction because an observer in the bus sees the car moving forward (past the
bus) at 8 kmh-1.
• The velocity of the car relative to the bus is the velocity of the car minus the
velocity of the bus. We can write this as a simple equation:
vcb = vc - vb Where c = car
~ ~ ~ b = bus
The vector subtraction is shown below:
8 kmh-1 12 kmh-1
20 kmh-1

20. RELATIVE VELOCITY in 2D
Consider the example of an aircraft carrier which is floating down a canal with a
speed of 2 ms-1. An officer walks across the flat deck of the carrier at right angles to
the sides of the vessel with a speed of 2 ms-1 as shown.
N The bank
N
2 ms-1 2 ms-1

21. RELATIVE VELOCITY in 2D
Consider the example of an aircraft carrier which is floating down a canal with a
speed of 2 ms-1. An officer walks across the flat deck of the carrier at right angles to
the sides of the vessel with a speed of 2 ms-1 as shown.
N The bank
N N
2 ms-1
2 ms-1 2 ms-1 -1
2 ms

22. RELATIVE VELOCITY in 2D
Consider the example of an aircraft carrier which is floating down a canal with a
speed of 2 ms-1. An officer walks across the flat deck of the carrier at right angles to
the sides of the vessel with a speed of 2 ms-1 as shown.
N The bank
2 ms-1
N N N
2 ms-1
2 ms-1 2 ms-1ms-1
2 ms-1
2

23. RELATIVE VELOCITY in 2D
Consider the example of an aircraft carrier which is floating down a canal with a
speed of 2 ms-1. An officer walks across the flat deck of the carrier at right angles to
the sides of the vessel with a speed of 2 ms-1 as shown.
N The bank
2 ms-1
2 ms-1
N N N N
2 ms-1
2 ms-1 2 ms-1ms-1 -1
2 ms-1ms
2 2

24. RELATIVE VELOCITY in 2D
Consider the example of an aircraft carrier which is floating down a canal with a
speed of 2 ms-1. An officer walks across the flat deck of the carrier at right angles to
the sides of the vessel with a speed of 2 ms-1 as shown.
N The bank
2 ms-1
2 ms-1
2 ms-1
N N N N N
2 ms-1
2 ms-1 2 ms-1ms-1ms-1
2 ms-1ms-1
2 2 2

25. RELATIVE VELOCITY in 2D
Consider the example of an aircraft carrier which is floating down a canal with a
speed of 2 ms-1. An officer walks across the flat deck of the carrier at right angles to
the sides of the vessel with a speed of 2 ms-1 as shown.
N The bank
2 ms-1
2 ms-1
2 ms-1
2 ms-1
N N N N N N
2 ms-1
2 ms-1 2 ms-1ms-1ms-1 -1
2 ms-1ms-1ms
2 2 2 2

26. RELATIVE VELOCITY in 2D
Consider the example of an aircraft carrier which is floating down a canal with a
speed of 2 ms-1. An officer walks across the flat deck of the carrier at right angles to
the sides of the vessel with a speed of 2 ms-1 as shown.
N The bank
2 ms-1
2 ms-1
2 ms-1
2 ms-1
2 ms-1
N N N N N N N
2 ms-1
2 ms-1 2 ms-1ms-1ms-1ms-1
2 ms-1ms-1ms-1
2 2 2 2 2

27. RELATIVE VELOCITY in 2D
Consider the example of an aircraft carrier which is floating down a canal with a
speed of 2 ms-1. An officer walks across the flat deck of the carrier at right angles to
the sides of the vessel with a speed of 2 ms-1 as shown.
N The bank
2 ms-1
2 ms-1
2 ms-1
2 ms-1
2 ms-1
N N N N N N N
2 ms-1
2 ms-1 2 ms-1ms-1ms-1ms-1
2 ms-1ms-1ms-1
2 2 2 2 2
The officer’s velocity relative to the moving object (the boat) is 2 ms-1 North
The velocity of the moving object is 2 ms-1 East
The officer’s velocity is a combination of these two velocities.

28. The officer’s direction is North - East.
The officer’s velocity can be calculated by adding the two velocity vectors:

29. The officer’s direction is North - East.
The officer’s velocity can be calculated by adding the two velocity vectors:
The officer’s velocity relative to the moving object

30. The officer’s direction is North - East.
The officer’s velocity can be calculated by adding the two velocity vectors:
The officer’s velocity relative to the moving object
2 ms-1

31. The officer’s direction is North - East.
The officer’s velocity can be calculated by adding the two velocity vectors:
The officer’s velocity relative to the moving object
2 ms-1
The velocity of the moving object

32. The officer’s direction is North - East.
The officer’s velocity can be calculated by adding the two velocity vectors:
The officer’s velocity relative to the moving object
2 ms-1
The velocity of the moving object
2 ms-1

33. The officer’s direction is North - East.
The officer’s velocity can be calculated by adding the two velocity vectors:
The officer’s velocity relative to the moving object
2 ms-1
The velocity of the moving object
2 ms-1
2 ms-1

34. The officer’s direction is North - East.
The officer’s velocity can be calculated by adding the two velocity vectors:
The officer’s velocity relative to the moving object
2 ms-1
The velocity of the moving object
2 ms-1
2 ms-1
2 ms-1

35. The officer’s direction is North - East.
The officer’s velocity can be calculated by adding the two velocity vectors:
The officer’s velocity relative to the moving object
2 ms-1
The velocity of the moving object
2 ms-1
2 ms-1
2 ms-1
The officer’s velocity, v

36. The officer’s direction is North - East.
The officer’s velocity can be calculated by adding the two velocity vectors:
The officer’s velocity relative to the moving object
2 ms-1
The velocity of the moving object
2 ms-1
2 ms-1
2 ms-1
v = 22 + 22
The officer’s velocity, v

37. The officer’s direction is North - East.
The officer’s velocity can be calculated by adding the two velocity vectors:
The officer’s velocity relative to the moving object
2 ms-1
The velocity of the moving object
2 ms-1
2 ms-1
2 ms-1
v = 22 + 22
The officer’s velocity, v
= 2.8 ms-1 NE

38. Intuitively:
The officer is moving at 2 ms-1 North at the same time as he is moving at 2 ms-1
East. It follows that the officer’s velocity (relative to the ground) is the sum of these
2 motions.
In other words: “The velocity of the officer relative to the ground is equal to the
velocity of the officer relative to the boat plus the velocity of the boat relative to the
ground”
We can write this as an equation:
Where o = officer
vog = vob + vb
g b = boat
~ ~ ~
g = ground
PROCESS FOR PROBLEM-SOLVING
1. Read the question carefully ---> Underline relevant information.
2. Assign a variable (a letter) to each object in the system. Eg. let g = the ground
3. Construct the vector symbol equation that relates the relative velocities to each
other.
4. Rearrange the equation if required.
5. Draw the vector diagram from the equation.
6. Solve for the unknown quantity.

39. EXAMPLES
1. A plane flies East with an air speed of 500 kmh-1 (this is the velocity of the plane
relative to the air). There is a wind blowing in the opposite direction at a speed of
30 kmh-1. Calculate the velocity of the plane relative to the ground by answering
the questions below:
(a) Give letters to symbolise the air, plane and ground.
(b) Write a vector equation that relates the 3 velocities. Below each vector symbol
sketch the vector showing size and direction.
(c) Perform the vector calculation to get your answer.

40. 2. A plane flies at 60 ms-1 due North, relative to the ground. A 20 ms-1 Northwest wind
is blowing (i.e. air coming from the Northwest).
Find:
(a) the airspeed of the plane
________________________________________________________________
________________________________________________________________
________________________________________________________________
(b) the direction that the pilot has to aim the plane to achieve his 20 ms-1 velocity
relative to the ground.
________________________________________________________________
________________________________________________________________
________________________________________________________________

41. 3. A ferry crosses a 500 m wide river in 1.5 minutes. The ferry must travel across the
river in a straight line from one ferry landing to the other as shown in the diagram
(below). The ferry has a water speed of 10 ms-1. Find the speed of the current.
Ferry landing
Current
500 m
Ferry
Ferry landing
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
Ex.8B Q.1 to 6

42. 12 PHYSICS RELATIVE VELOCITY ASSIGNMENT Name
1. A man swims in a river which has a current of 2.5 ms-1. He finds that he can swim
upstream, against the current, at 1.5 ms-1 relative to the river bank.
bank
river current
bank
(a) What is his velocity relative to the water?
________________________________________________________________
(b) He then swims downstream with the same effort. What is his velocity relative to
the bank?
________________________________________________________________
________________________________________________________________
(c) The man then swims (at 4 ms-1 relative to the water) so that he faces directly
across the river. As he swims across, the current takes him downstream a little.
Draw a vector velocity triangle to show this.

43. 2. A small plane is flying West at 60 ms-1. A 20 ms-1 South wind springs up.
(a) Sketch this situation
(b) The pilot has to continue at 60 ms-1 Westwards despite the wind. Sketch a
vector triangle which shows what the pilot should do.
(c) Calculate the direction of the plane’s air speed (i.e. its speed relative to the air).
________________________________________________________________
________________________________________________________________
(d) Calculate the magnitude of the plane’s air speed.
________________________________________________________________
________________________________________________________________
________________________________________________________________

44. 3. A man rows his boat North directly across a river at 2.0 ms-1. A current starts to
flow East at 1.3 ms-1. bank
(a) Sketch a vector triangle of
the velocities. Label each vector. river current
Boat
bank
(b) Write a vector equation for the velocities.
________________________________________________________________
(c) Calculate the magnitude of the boat’s velocity relative to the bank.
________________________________________________________________
________________________________________________________________
(d) Calculate the direction of the boat’s velocity relative to the bank.
________________________________________________________________
________________________________________________________________
________________________________________________________________