# 17.1 Displacement and distance

 Page 1 / 1

## Displacement and distance

Displacement

Displacement is the change in an object's position.

The displacement of an object is defined as its change in position (final position minus initial position). Displacement has a magnitude and direction and is therefore a vector. For example, if the initial position of a car is ${x}_{i}$ and it moves to a final position of ${x}_{f}$ , then the displacement is:

${x}_{f}-{x}_{i}$

However, subtracting an initial quantity from a final quantity happens often in Physics, so we use the shortcut $\Delta$ to mean final - initial . Therefore, displacement can be written:

$\Delta x={x}_{f}-{x}_{i}$
The symbol $\Delta$ is read out as delta . $\Delta$ is a letter of the Greek alphabet and is used in Mathematics and Science to indicate a change in a certain quantity, or a final value minus an initial value. For example, $\Delta x$ means change in $x$ while $\Delta t$ means change in $t$ .
The words initial and final will be used very often in Physics. Initial will always refer to something that happened earlier in time and final will always refer to something that happened later in time. It will often happen that the final value is smaller than the initial value, such that the difference is negative. This is ok!

Displacement does not depend on the path travelled, but only on the initial and final positions ( [link] ). We use the word distance to describe how far an object travels along a particular path. Distance is the actual distance that was covered. Distance (symbol $D$ ) does not have a direction, so it is a scalar. Displacement is the shortest distance from the starting point to the endpoint – from the school to the shop in the figure. Displacement has direction and is therefore a vector.

[link] shows the five houses we discussed earlier. Jack walks to school, but instead of walking straight to school, he decided to walk to his friend Joel's house first to fetch him so that they can walk to school together. Jack covers a distance of $400\phantom{\rule{2pt}{0ex}}\mathrm{m}$ to Joel's house and another $500\phantom{\rule{2pt}{0ex}}\mathrm{m}$ to school. He covers a distance of $900\phantom{\rule{2pt}{0ex}}\mathrm{m}$ . His displacement, however, is only $100\phantom{\rule{2pt}{0ex}}\mathrm{m}$ towards the school. This is because displacement only looks at the starting position (his house) and the end position (the school). It does not depend on the path he travelled.

To calculate his distance and displacement, we need to choose a reference point and a direction. Let's choose Jack's house as the reference point, and towards Joel's house as the positive direction (which means that towards the school is negative). We would do the calculations as follows:

$\begin{array}{ccc}\hfill \mathrm{Distance}\left(\mathrm{D}\right)& =& \mathrm{path}\phantom{\rule{3.33333pt}{0ex}}\mathrm{travelled}\hfill \\ & =& 400\phantom{\rule{4pt}{0ex}}\mathrm{m}+500\phantom{\rule{4pt}{0ex}}\mathrm{m}\hfill \\ & =& 900\phantom{\rule{4pt}{0ex}}\mathrm{m}\hfill \end{array}$
$\begin{array}{ccc}\hfill \mathrm{Displacement}\left(\Delta \mathrm{x}\right)& =& {x}_{f}\phantom{\rule{3.33333pt}{0ex}}-\phantom{\rule{3.33333pt}{0ex}}{x}_{i}\hfill \\ & =& -100\phantom{\rule{4pt}{0ex}}\mathrm{m}+0\phantom{\rule{4pt}{0ex}}\mathrm{m}\hfill \\ & =& -100\phantom{\rule{4pt}{0ex}}\mathrm{m}\hfill \end{array}$

You may also see $\mathrm{d}$ used for distance. We will use $\mathrm{D}$ in this book, but you may see $\mathrm{d}$ used in other books.

Joel walks to school with Jack and after school walks back home. What is Joel's displacement and what distance did he cover? For this calculation we use Joel's house as the reference point. Let's take towards the school as the positive direction.

$\begin{array}{ccc}\hfill \mathrm{Distance}\left(\mathrm{D}\right)& =& \mathrm{path}\phantom{\rule{3.33333pt}{0ex}}\mathrm{travelled}\hfill \\ & =& 500\phantom{\rule{4pt}{0ex}}\mathrm{m}+500\phantom{\rule{4pt}{0ex}}\mathrm{m}\hfill \\ & =& 1000\phantom{\rule{4pt}{0ex}}\mathrm{m}\hfill \end{array}$
$\begin{array}{ccc}\hfill \mathrm{Displacement}\left(\Delta \mathrm{x}\right)& =& {x}_{f}\phantom{\rule{3.33333pt}{0ex}}-\phantom{\rule{3.33333pt}{0ex}}{x}_{i}\hfill \\ & =& 0\phantom{\rule{4pt}{0ex}}\mathrm{m}+0\phantom{\rule{4pt}{0ex}}\mathrm{m}\hfill \\ & =& 0\phantom{\rule{4pt}{0ex}}\mathrm{m}\hfill \end{array}$

It is possible to have a displacement of $0\phantom{\rule{2pt}{0ex}}\mathrm{m}$ and a distance that is not $0\phantom{\rule{2pt}{0ex}}\mathrm{m}$ . This happens when an object completes a round trip back to its original position, like an athlete running around a track.

## Interpreting direction

Very often in calculations you will get a negative answer. For example, Jack's displacement in the example above, is calculated as $-100\phantom{\rule{2pt}{0ex}}\mathrm{m}$ . The minus sign in front of the answer means that his displacement is $100\phantom{\rule{2pt}{0ex}}\mathrm{m}$ in the opposite direction (opposite to the direction chosen as positive in the beginning of the question). When we start a calculation we choose a frame of reference and a positive direction. In the first example above, the reference point is Jack's house and the positive direction is towards Joel's house. Therefore Jack's displacement is $100\phantom{\rule{2pt}{0ex}}\mathrm{m}$ towards the school. Notice that distance has no direction, but displacement has.

## Differences between distance and displacement

The differences between distance and displacement can be summarised as:

 Distance Displacement 1. depends on the path 1. independent of path taken 2. always positive 2. can be positive or negative 3. is a scalar 3. is a vector 4. does not have a direction 4. has a direction

## Point of reference

1. Jill walks to Joan's house and then to school, what is her distance and displacement?
2. John walks to Joan's house and then to school, what is his distance and displacement?
3. Jack walks to the shop and then to school, what is his distance and displacement?
4. What reference point did you use for each of the above questions?
2. You stand at the front door of your house (displacement, $\Delta x=0\phantom{\rule{2pt}{0ex}}\mathrm{m}$ ). The street is $10\phantom{\rule{2pt}{0ex}}\mathrm{m}$ away from the front door. You walk to the street and back again.
1. What is the distance you have walked?
2. What is your final displacement?
3. Is displacement a vector or a scalar? Give a reason for your answer.

Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
The fundamental frequency of a sonometer wire streached by a load of relative density 's'are n¹ and n² when the load is in air and completly immersed in water respectively then the lation n²/na is
Properties of longitudinal waves