4.1.1 Scalars and vectors
Specification
Nature of scalars and vectors.
Examples should include:
velocity/speed, mass, force/weight, acceleration, displacement/distance.
Addition of vectors by calculation or scale drawing.
Calculations will be limited to two vectors at right angles. Scale drawings may involve
vectors at angles other than 90°.
Vectors and scalars
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Displacement and velocity
A runner completes one lap of
an athletics track.
What distance has she run?
400 m
What is her final displacement?
If she ends up exactly where she
started, her displacement from
her starting position is zero.
What is her average velocity for
the lap, and how does it
compare to her average speed?
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Displacement vectors
Harry and Sally are exploring the desert. They need to
reach an oasis, but choose to take different routes.


Harry travels due north,
then due east.
N
Sally simply travels in a
straight line to the oasis.
When Harry met Sally at the oasis, they had travelled
different distances. However, because they both reached
the same destination from the same starting point, their
overall displacements were the same.
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Scalar or vector?
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4.1.1 Scalars and vectors
To find the size and direction of the
force you can use a scale drawing
• Eg scale 10mm = 1 N
•
3N
•
4N
• FLIP THE TRIANGLE!!!!!!!
•
•
5N
3N
•
4N
You can also find the size and direction of
the resultant force by calculation
• Use Pythagoras's theorem!!!!
• F2 = 32 + 42
• F2 = 25
• F = 5N
Size of angle
• Tan θ =3/4 (SOHCAHTOA)
• θ = 37o
•
•
•
3N (opp)
θ
4N (adj)
Calculating a resultant
When adding two perpendicular vectors, it is often necessary
to calculate the exact magnitude and direction of the
resultant vector. This requires the use of Pythagoras’
theorem, and trigonometry.
For example, what is the resultant vector of a vertical
displacement of 3 km and a horizontal displacement of 4 km?
R
4 km
θ
magnitude:
direction:
R2 = 32 + 42
tan θ = 4/3
R = √ 32 + 42
= √ 25
θ = tan-1(4/3)
= 53°
= 5 km
3 km
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Worked examples
1.1.5
1.1.5
A lorry’s load has a weight of 40 000 N and it is being pulled along by a
horizontal force of 15 000 N. What is the resultant of these two forces?
With this type of calculation it is often easier if you
sketch a rectangle using two triangles as in the Figure.
Here, the rectangle shows the two forces acting at right
angles to one another, and the diagonal represents the
resultant force.
Using Pythagoras’s theorem gives:
resultant2 = (4.0 × 104)2 + (1.5 × 104)2
= 16.0 × 108 + 2.25 × 108
= 18.25 × 108 so
resultant = 4.27 × 104 N
Worked examples
1.1.5 (cont.)
1.1.5 (cont.)
When working out such calculations, you need to be careful if you are using
powers of 10. Always check that the length of the hypotenuse is realistic.
This has given you the magnitude of the resultant force, but to complete the
calculation its direction is also required. Since the adjacent side is given by
hypotenuse × cosθ we get:
cosθ =
4.00 10 4
4.27 10 4
= 0.9368 and hence θ = 20.5°
Note: Worst case scenario
Much worst case scenario
A raindrop falls at a constant velocity of 1.8 ms-1. If a horizontal
wind of 1.4ms-1 is blowing, calculate the magnitude and
direction of the resultant velocity of the raindrop by
(1) a scale diagram
(2) calculation
Resultant
velocity
1.8ms-1
x2 = 1.82 + 1.42
x = √1.82 + 1.42
x = 2.28
1.4ms-1
1.8ms-1
Tanθ = 1.4/1.8
θ
Tanθ = 0.7778
SOH CAH TOA
θ = 37.90
1.4ms-1
Scale:
1cm = 0.2 ms-1
1.8 ms-1
Resultant
velocity
Measure length of resultant
velocity = 11.5 cm
1cm = 0.2 ms-1
11.5 x 0.2 = 2.3 ms-1
1.4 ms-1
Resultant vectors
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On a windy day, a parachutist of mass 85 kg
jumps from an aeroplane.
Distinguish between speed and velocity.
On a windy day, a parachutist of mass 85 kg
jumps from an aeroplane.
As the parachutist falls, the wind is moving
him towards the right of the diagram, at a
horizontal velocity of 6.3 m/s.
Draw a vector diagram to determine
graphically the size and
direction of the resultant velocity of the
parachutist.
size = .......................................................
direction = .......................................................
[4]
On a windy day, a parachutist of mass 85 kg
jumps from an aeroplane.
Calculate the kinetic energy of the parachutist.
kinetic energy = .................................................
[3]
Vehicle A exerts 4000N
Vehicle B exerts 2000N