Mechanics Tutorial 1  Vectors and Scalars
acceleration;
force;
velocity.
energy;
temperature.
Vector

Scalar

Unit

Displacement 
Distance 
Metres
(m) 
Velocity 
Speed 
Metres per second (m/s) 
Acceleration 

Metres
per second^{2} (m/s^{2}) 
Momentum 

Newton
seconds (Ns) 
Force 

Newtons
(N) 

Work
, Energy 
Joules
(J) 

Voltage 
Volts
(V) 

Temperature 
Degrees
Celsius (^{o}C) 

Frequency 
Hertz
(Hz) 
Notice
that the units are the same,
regardless of whether they are vectors or scalars.
Note:
Some vectors can be used as scalar quantities.
There are some scalar quantities, e.g. temperature, that have no vector equivalent.
The shorthand for metres per second is either written ms^{1 }or m/s. Either is acceptable.
Work is the product of two vectors (Work = Force × distance moved in direction of force) but it is a scalar.
If the force vectors of 3N and 4N are in the same direction, they simply add together.
The
heavy arrow indicates the resultant
force.
If the vectors are in opposite directions, we subtract.
We can see that the resultant is now just 1 N.
If the two vectors are at 90^{o} use Pythagoras’ Theorem.
Resultant^{2} = 3^{2}
+ 4^{2} = 9 + 16 = 25.
\ Resultant = Ö(25) = 5 N
To
work out the angle we use the tan
function:
tan
q
= ¾ = 0.75
Þ q
=
tan^{1}(0.75) = 36.9^{o}
We can also add vectors that are not at right angles:
In the picture above we can see the resultant of two forces that are not at right angles. We can show that they make a vector triangle by moving Force 1:
Alternatively we can use a parallelogram of forces as shown below:
The resultant can be worked out by accurate drawing. Or you can use the cosine rule:
At AS level, you would only have to add vectors at 90 degrees to each other. If there were vectors as shown above, then you would normally be expected to use accurate drawing. The question would tell you to do accurate drawing, although I am sure that if you got the right answer from the cosine rule, you would be awarded full credit.
What are the resultants of these vectors? 

Question 2  What are the angles that the resultants make to the vertical in the previous question? 
We can resolve any vector into two components at 90^{o} to each other. They are called the vertical and the horizontal components.
F_{x}
= F
cos
q
F_{y} = F sin q
Consider a car going up a hill.
The angle of the hill is q degrees. We must note that the weight (given by the mass in kilograms × acceleration due to gravity) is always pointing vertically down. Acceleration due to gravity can be taken as 9.8 m s^{2} and the force of gravity is 9.8 N/kg. We can resolve the vectors, remembering that the weight acting vertically is the resultant force.
Remember:
It is depressing how many students write weight in kilograms. Watch out for this bear trap!
The car has a mass of 1100 kg, and the angle is 10^{o}. Calculate: (a) the weight of the car, (b) the force on the road, (c) the downhill force. 