Our Dynamic Universe: Scalars and vectors
Our Dynamic Universe
SCALARS AND VECTORS
MATHEMATICAL METHODS
In this section, we will consider the idea of motion, how to combine scalar and vector quantities, and
use graphical and mathematical methods when dealing with scalar and vector quantities. Through this
course you will meet both scalar and vector quantities.
You must remember to calculate both magnitude and direction.
Know the resultant direction from start
to finish.
A body or an object is said to
be in motion when it changes
its position. We can analyse
an object’s displacement,
velocity and acceleration
during its change of position.
These are vector quantities
so both magnitude (size) and
direction must be stated for
each. A scalar quantity has
no direction to give.
Scalar (Magnitude only)
Vector (Magnitude and direction)
Time
Energy
Mass
Distance
Displacement
Speed
Velocity
Acceleration
Momentum
Force
Impulse
Displacement
finish
(shop)
Displacement is the distance
travelled in a certain direction
from the start point to the
finish point in a straight line.
Velocity
Velocity is the rate of change
of displacement. Velocity is
the displacement covered in a
certain time. Direction has to
be stated.
t
en
em
lac
sp
di
Acceleration
Acceleration is the rate
of change of velocity.
Acceleration is the change
of velocity done in a certain
time. Acceleration takes place
in a certain direction.
start
(school)
Different paths, same displacement
SCALE DRAWINGS
DON’T FORGET
When drawing your scale
diagram keep it large to
reduce the uncertainty in
the answer.
Vector quantities can be represented with a line drawn to a scale. An arrowhead must be
added to show which direction the line represents. Remember these rules to add vector
quantities together:
1 Choose and write down a scale.
2 Add vectors drawn ‘head to tail’.
3 Draw the resultant from ‘start to finish’.
4 Measure both the magnitude and the direction.
An arrow has magnitude and direction
8
BRP_CfEHigherPhysics_Sample.indd 8-9
Right angled triangles
1 Use the Pythagorean theorem to calculate the resultant magnitude:
SAMPLE PAGES – CFE HIGHER PHYSICS
MOTION
When adding vector quantities together or resolving a vector into components, in certain
diagrams you may find you can use mathematical methods as an alternative to scale diagrams.
c2 = a2 + b2
B
(hypotenuse)
c
a
(opposite)
θ
A
b
(adjacent)
C
2 Use trigonometry to calculate the direction:
opposite
tan θ = adjacent
opposite
sin θ = hypotenuse
adjacent
cos θ = hypotenuse
Acute or obtuse angled triangles
1 Use the Sine rule to calculate the size of an angle or length of a side when you have
two sides and an angle or two angles and a side:
a
b
c
= sin B = sin C
sin A
2 Use the Cosine rule to calculate the length of a side or size of an angle when you have
two sides and an angle or three sides:
a2 = b2 + c2 – 2bc cos A
THINGS TO DO AND THINK ABOUT
ONLINE
For more on vectors, head
to www.brightredbooks.net
ONLINE TEST
When an object falls to the ground its
velocity is said to be increasing and it
has an acceleration which is vertically
downwards towards the centre of the
earth. An object in orbit is usually
said to have a constant orbital speed
but gravity is of course changing its
direction to keep it in orbit. This object
is also said to be accelerating towards
the earth. How can an object with
constant speed also have acceleration?
Test your knowledge of
this topic at
www.brightredbooks.net
DON’T FORGET
Acceleration is a vector
quantity.
Orbits of satellites accelerating
9
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Our Dynamic Universe: Displacement and velocity
Our Dynamic Universe
DISPLACEMENT AND VELOCITY
SPEED AND VELOCITY
Speed is a scalar quantity. Speed is the distance travelled in unit time.
speed =
In this section, we will consider the vector nature of displacement and velocity. We will also learn how
to use vector rules to add these quantities.
The symbol for distance is d. Distance is defined by a number and its unit, the metre, m.
e.g. ‘The length of the school laboratory is 8 m’.
Displacement is a vector quantity. e.g. ‘An explorer walks 20 km due North from base
camp’. The symbol for displacement is s.
Example: 1
A pupil travels East 10 m along the school corridor but is sent back 7 m for running!
distance travelled, d = 10 + 7 = 17 m
displacement is 10 + (–7) = 3 m East.
10
N
–7
If this pattern is repeated, we can predict the pupil’s displacement will still only be 6 m East.
Example: 2
3
A car drives 4 km North then drives 3 km West.
What is the resultant displacement?
Using Pythagoras and trig:
x2 = 42 + 32
N
x = 5 km
x
tan θ = ¾
θ = 37º
θ
displacement, s = 5 km @ 37º to the West of North.
Use a ruler and protractor to check this by scale measurement.
Example: 3
A runner goes round the race track three times. If the track has a length of 220 m,
what is the runner’s distance travelled and displacement?
The magnitude of the
displacement is different
from the distance when an
object changes direction.
finish start
If we ignore direction when
calculating vector quantities,
we will usually get the
wrong result!
distance travelled
d = 3 × 220 = 660 m.
BRP_CfEHigherPhysics_Sample.indd 52-53
Velocity is a vector quantity. Velocity is the displacement per unit time.
velocity =
displacement
s
or v = t
time
Velocity has magnitude and direction.
e.g. ‘The car was travelling at 30 m s–1 from Glasgow to Edinburgh’ refers to a velocity.
Example:
DON’T FORGET
A sailor sets his boat on a heading of North at 5 m s–1
through the sea. The tide is moving at 2 m s–1 in a
South-East direction.
2ms –1
Find the boat’s resultant velocity over the ground.
Scale: 1 cm ≡ 1 m s–1
Head-to-tail
diagram:
2
If a bearing has been
used for the direction,
000° is North.
N
DON’T FORGET
boat
5ms –1
… the direction, or you could
lose half the marks!
Resultant from
start to finish
5
θ
x
DON’T FORGET
… these vector techniques;
they will be used to add
force vectors soon.
VIDEO LINK
Measure resultant, x = 3.9 cm
Measure θ = 22°
Resultant velocity, v = 3.9 m s–1 @ 022°
THINGS TO DO AND THINK ABOUT
The use of s for displacement is from the Latin word for space: spatium, first used in
Galileo’s Discourses on Two New Sciences in 1640.
There are many online
video clips about
displacement and velocity
vectors. View a selection at
www.brightredbooks.net
ONLINE TEST
Test your knowledge
of this topic at
www.brightredbooks.net
Use your knowledge of vectors to explain to a friend how your velocity changes when:
DON’T FORGET
52
4
SAMPLE PAGES – CFE HIGHER PHYSICS
A body in motion will often change its direction. The distance travelled from start to
finish can vary with the routes taken but displacement remains as the distance travelled
in a straight line in the direction from start to finish.
DON’T FORGET
Speed only has magnitude, no direction.
e.g. ‘The car was going faster than 30 m s–1’ refers to a speed.
DISTANCE AND DISPLACEMENT
3
distance
time
•
•
•
you walk on a conveyor belt
a plane heads into a crosswind
you sail against the tide.
displacement
s = 0 m.
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