After distance and displacement that we have seen earlier, I am now going to talk about speed and velocity. They are quite different and to understand them it is important to understand clearly the difference distance and displacement before going forward.
Speed
Speed is a scalar quantity and the unit is the metre per second (m s-1)
A boy moves from point A and walks a distance of 50 m to the point B in a time of 20 s.
Firstly what is the distance travelled in 1 second?
Yes the boy would walk a distance of 2.5 m in a time of 1 second.
The distance travelled in a time of 1 second is called the speed.
The equation to calculate the speed is shown below.
Can you determine the speed of the boy?
Hence whenever speed is calculated the distance should be used.
Velocity
Velocity is a vector quantity and the unit is also the metre per second(m s-1)
Since it is a vector quantity then when it is determined direction is important and hence displacement should be used.
A boy moves from point A to point B performing a displacement of 50 m towards the right as shown in the diagram below.
The following equation is used to determine the velocity.
Hence the velocity of the boy can be calculated as shown below.
Please note the difference between the two. For the speed the distance travelled is used while for the velocity the displacement is used.
Distance and displacement are two quantities that are used to describe motion. We are going to have a look at both of them and see how to differentiate between the two.
Distance
Distance is a scalar quantity and the SI unit is the Metre (m)
Example 1
Now suppose a person moves from an initial position of A and travels to a final position of B. The line connecting the two points is the path that the person would take. Now the distance travelled is the length of the path taken by the person from the initial position to the final position.
Example 2
Now if the path is not a straight line like in the diagram shown above then the distance travelled is the length of the path from A to B through all the turns as measured with a measuring tape.
Example 3
The next example is shown above. The person moves from A to D. The path is a line moving from A to B to C and then finally to D. Then the distance travelled is the length of the path AB+ the length of the path BC + the length of the path CD
Displacement
Displacement is a vector quantity and the SI unit is the metre (m)
The displacement is a vector representing the shortest path from the initial position to the final position. It is thus a vector quantity. Let us have a look at the different motion above and see how the displacement is determined and how it is different from the distance.
Example 4
As we have seen in example 1 above the person moves from the initial position A to the final position B.
As you can see the person moves from the initial position A to the final position B and the vector representing this motion is as shown. And the displacement is represented by this vector. Hence the displacement will have both a magnitude which would be the length of the vector and a direction which would be the direction of the vector.
For example 4 above we can say that the displacement is 10 m to the east.
Displacement = 10 m to the east
or Displacement = 10 m to the right
As you can see the displacement would have both a magnitude and a direction since it is a vector quantity.
Example 5
This is the same motion that we used in example 2. The person moves from the initial position A and travels until he reaches the final position B. As you can see on the diagram above there is a vector from the initial position A to the final position B. This vector represent the displacement of the person where the length of the vector is its magnitude and the direction of the vector the direction of the displacement.
In this case the displacement of the person can be said to be
Displacement = 10 m 15o clockwise from the horizontal,
Where the line AC represent the easterly direction,
Example 6
Now let us look at the same motion example that the person did in example 3. The person walks from A to B to C and finally arrives at D. You can see on the diagram below that I have drawn a vector AD that starts from the point A and ends at the point D. This vector is the displacement vector and as you can see it has a magnitude and a direction.
We can say for example that the displacement is as shown
Displacement = 15 m 150 clockwise from the horizontal
As you have seen displacement and distance are two quantities easy to understand and to differentiate but as usual if you have any question you can leave them here and I would answer then as soon as possible.
This is the second part of a series of post on finding uncertainty in derived quantities. You can find the index here and the part on addition and subtraction here.
Now very often when you performed an experiment you meet quantities that need to be processed to obtain a derived quantity such as g, the acceleration due to gravity, or any other derived quantities.
You have already seen the first part on addition and subtraction and now you will see how to do it for multiplication and division.
If in an experiment is performed and the following quantities are measured with their uncertainties.
A = 10.2+-0.2 cm
B = 5.4 +-0.4 cm
Now if you need to process these quantities to find AB and A/B with their uncertainties how would you do it?
Uncertainty in multiplication
How would you determine the value of AB and its uncertainty?
You will have to calculate the value of AB first.
AB = 10.2 *5.4 = 55.o8 =55 (2 sf )
To determine the uncertainty in AB you will have to use the equation below.
Rearranging the equation will give you
Hence AB = 55+-5 cm2
Remember the uncertainty in A and B are to 1 sf hence the uncertainty in AB must be given to 1 sf.
Uncertainty in division
You are now going to determine the value of A/B and its uncertainty.
You will have to determine the value of A/B first.
A/B = 10.2 /5.4 =1.888 = 1.9
To determine the uncertainty of A/B you will determine the equation below
Rearranging the equation will give you
Remember the uncertainty in A/B is given to 1 sf since the uncertainties in A and B are given to 1 sf.
Hence A/B = 1.9 +-0.2
With these two formula you can thus determine the uncertainty in any derived quantities that involves multiplication and division.
Now you can move to the next part where the method to determine uncertainty of derived quantities where powers, square root, etc are involved.
In physics all quantity that is measured come with its uncertainty. This is because the instrument, the experiment or the person performing the measurement is subjected to some limitations.
For example
Someone that is measuring a length with a metre rule cannot measure the length to a precision that is better than one mm. So a length that is measured with a metre rule will have an uncertainty of +-o.1 c m.
The limitation can either be due to the experiment itself, the apparatus, or the experimenter. Someone measuring the height of rebound will measure it with an uncertainty of +- 0.5 cm. Someone estimating the mass of an object can measure it with an uncertainty of +-0.5 kg. A voltmeter will give a reading with a percentage uncertainty of 1%.
[As you may have guess from 2 above not all measurements that are performed will have these uncertainty. These are for example only.]
As you can see all quantities that are measured will have uncertainties. And as a result any other quantities that are derived from the measured quantities will have their uncertainties.
We are going to see how to determine the uncertainty when the following operations are performed.
As we have seen in the last post there are 7 fundamental base units. From these 7 fundamental base units all the other units can be derived.
In fact if a quantity is not a basic quantity then it must certainly be a derived quantity and its unit a derived unit. The derived unit is based on one or more of the 8 fundamental base units. You must keep in mind that some quantities have units of their own based on famous scientists such as Joule (J) for energy or Newton (N) for force. For such quantities the unit can be used or the derived unit can be used.
We are now going to see several derived quantities and their derived units and how these derived units are obtained.
We are now going to see how these derived units are obtained. Once you know how to determine the derived units of common quantities like volume, force, etc then later on you will be able to determine the derived quantity for any other quantity.
So what is the method to obtain the derived unit of a quantity?
Example 1
We are going to start with a simple one: the derived unit for volume.
In order to deduce the derived unit of a quantity you need a formula to calculate the quantity. The formula must contain that you know the units in term of based unit.
Volume = length * width * height
Unit of [volume] = unit of [length * width * height]
= unit of [length] * unit of [width] * unit of [height]
= m * m *m
= m3
Example 2
What is the derived unit for acceleration?
In order to deduce the derived unit for acceleration, you must know a formula to calculate acceleration such as
Acceleration = velocity / time
The formula must contain quantities that you know the units in term of base units.
If in the new formula, there is quantity that you do not know the derived unit then you will also have to know the formula to calculate that quantity.
Velocity = displacement / time
You can now rewrite the formula for acceleration to
Acceleration = velocity / time
= (displacement / time) /time
So
the unit of acceleration = unit of [velocity / time]
= unit of [(displacement / time) /time]
= unit of [(displacement / time)] / unit of time
= (m/s) /s
= m *s-1 * s-1 = m s-2
Was it easy?
Now let us have a look at another example to confirm your newly learned skills.
Example 3
What is the derived unit for pressure?
Have you worked it out?
Now you would remember that in order to find the derived unit you need to know a formula for the quantity in term of quantities that you already know the units.
Pressure = Force / surface are
Hence unit of pressure = unit of [force] / unit of [surface area]
= unit of [mass * acceleration] / unit of surface area]
= (kg * m/ s2)/m2 = (kg m * s-2 * m -2 = Kg m-1 s-2
Do the following questions and post the answers in the comment section. I will give the correct answers later on after a few of you have submitted your answers.
Find the derived units for the following quantities.
1. Work done
2. Kinetic energy
3. Volume
4. Density
As you progress through the course you will meet more quantities that you would be able to derive the units.
Units are the third aspect that is important when measuring in Physics. It can be the mistake that will ruin the whole measurement process.
Let us recap.
You need to identify the quantity that is to be measured eg length of the book.
you need to use the correct instrument to measure the quantity. In this case the 30 cm ruler.
Lastly you need to measure the length and write it with appropriate unit.
Imagine that you write the length of a 20 cm long book as 20 inches. Then it ruins the whole measurement process.
SI units
To avoid such mistakes scientist around the world have decided to use what is called the SI system of units. It is a standard that is used by all scientists such that when a particular quantity is being measures everyone uses the same units although the Americans and the British continue to use the old systems. This has resulted in costly mistakes and loss of human lives in the space industry.
Base units
The SI system is based on fundamental base units on which all other units can be derived.
The 7 fundamental base units are given in the table below.
Basic Quantity
Base unit
Symbol of base unit
Length
Metre
m
Time
Second
s
Mass
Kilogram
kg
Thermodynamic temperature
Kelvin
K
Luminous intensity
Candela
Cd
Electric current
Ampere
A
Amount of matter
Mole
Mol
These units are the 7 fundamental base units that are used in the SI system of units. Hence even though the quantities in the above table can be measured using other units, it is recommended that only these units are used.
The next part this series on derived units can be found here.
A physical quantity is a property of an object that can be measured with a measuring instrument.For example length, width, time, weight, etc.
It is important for you to be able to say which instrument is used to measure which physical quantity.The table below shows a few examples of physical quantities and the instruments used to measure them.
Physical quantityInstrument
Length Metre rule
Temperature Thermometer
Time Stopwatch
Volume Measuring cylinder
Of course as we go through the different chapters we will be able to introduce more physical quantities and their measuring instruments.
Basic and derived quantities
Quantities can be divided into two types :
1. Base quantities
2. Derived quantities
Basic quantities
Basic quantities are the fundamental quantities that are not related to each other and that are use to derive all other quantities.
There are seven basic quantities. They are
1. length
2. time
3. mass
4. Thermodynamic temperature
5. electric current
6. amount of substance
7. luminous intensity
Derived quantities
Derived quantities are just quantities that are derived from one or more basic quantities.
For example area is a derived quantity because it is derived from the basic quantity length.
Area = length * length
Volume is a derived quantities because it is derived from the basic quantity length.
volume = length * length * length
density is a derived quantity because it is derived from length and mass, two basic quantities.
density = mass/(length * length * length)
As you can see all other quantities apart from the 7 basic quantities are derived from the seven basic quantities and it can easily be shown as in the example above.
Can you show what are the basic quantities that are used to derive the following derived quantities?
a) force
b) kinetic energy
c) power
d) pressure
I would give the answer after a few of you would have give your answers.
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