# 0.2 Motion in one dimension  (Page 11/16)

 Page 11 / 16

To calculate the velocity of the taxi you need to calculate the gradient of the line at each second:

$\begin{array}{ccc}\hfill {v}_{1s}& =& \frac{\Delta x}{\Delta t}\hfill \\ & =& \frac{{x}_{f}-{x}_{i}}{{t}_{f}-{t}_{i}}\hfill \\ & =& \frac{5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}-0\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}}{1,5\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}-0,5\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}}\hfill \\ & =& 5\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}·{\mathrm{s}}^{-1}\hfill \end{array}$
$\begin{array}{ccc}\hfill {v}_{2s}& =& \frac{\Delta x}{\Delta t}\hfill \\ & =& \frac{{x}_{f}-{x}_{i}}{{t}_{f}-{t}_{i}}\hfill \\ & =& \frac{15\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}-5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}}{2,5\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}-1,5\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}}\hfill \\ & =& 10\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}·{\mathrm{s}}^{-1}\hfill \end{array}$
$\begin{array}{ccc}\hfill {v}_{3s}& =& \frac{\Delta x}{\Delta t}\hfill \\ & =& \frac{{x}_{f}-{x}_{i}}{{t}_{f}-{t}_{i}}\hfill \\ & =& \frac{30\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}-15\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}}{3,5\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}-2,5\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}}\hfill \\ & =& 15\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}·{\mathrm{s}}^{-1}\hfill \end{array}$

From these velocities, we can draw the velocity-time graph which forms a straight line.

The acceleration is the gradient of the $v$ vs. $t$ graph and can be calculated as follows:

$\begin{array}{ccc}\hfill a& =& \frac{\Delta v}{\Delta t}\hfill \\ & =& \frac{{v}_{f}-{v}_{i}}{{t}_{f}-{t}_{i}}\hfill \\ & =& \frac{15\phantom{\rule{0.166667em}{0ex}}\mathrm{m}·{\mathrm{s}}^{-1}-5\phantom{\rule{0.166667em}{0ex}}\mathrm{m}·{\mathrm{s}}^{-1}}{3\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}-1\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}}\hfill \\ & =& 5\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}·{\mathrm{s}}^{-2}\hfill \end{array}$

The acceleration does not change during the motion (the gradient stays constant). This is motion at constant or uniform acceleration.

The graphs for this situation are shown in [link] . Graphs for motion with a constant acceleration (a) position vs. time (b) velocity vs. time (c) acceleration vs. time.

## Velocity from acceleration vs. time graphs

Just as we used velocity vs. time graphs to find displacement, we can use acceleration vs. time graphs to find the velocity of an object at a given moment in time. We simply calculate the area under the acceleration vs. time graph, at a given time. In the graph below, showing an object at a constant positive acceleration, the increase in velocity of the object after 2 seconds corresponds to the shaded portion.

$\begin{array}{ccc}\hfill v=\mathrm{area}\phantom{\rule{4pt}{0ex}}\mathrm{of}\phantom{\rule{4pt}{0ex}}\mathrm{rectangle}& =& a×\Delta t\hfill \\ & =& 5\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}·{\mathrm{s}}^{-2}×2\phantom{\rule{4pt}{0ex}}\mathrm{s}\hfill \\ & =& 10\phantom{\rule{0.166667em}{0ex}}\mathrm{m}·{\mathrm{s}}^{-1}\hfill \end{array}$

The velocity of the object at $t=2\mathrm{s}$ is therefore $10\phantom{\rule{2pt}{0ex}}\mathrm{m}·\mathrm{s}{}^{-1}$ . This corresponds with the values obtained in [link] .

## Summary of graphs

The relation between graphs of position, velocity and acceleration as functions of time is summarised in [link] .

Often you will be required to describe the motion of an object that is presented as a graph of either position, velocity or acceleration as functions of time. The description of the motion represented by a graph should include the following (where possible):

1. whether the object is moving in the positive or negative direction
2. whether the object is at rest, moving at constant velocity or moving at constant positive acceleration (speeding up) or constant negative acceleration (slowing down)

You will also often be required to draw graphs based on a description of the motion in words or from a diagram. Remember that these are just different methods of presenting the same information. If you keep in mind the general shapes of the graphs for the different types of motion, there should not be any difficulty with explaining what is happening.

## Aim:

To measure the position and time during motion and to use that data to plot a “Position vs. Time" graph.

## Apparatus:

Trolley, ticker tape apparatus, tape, graph paper, ruler, ramp

## Method:

1. Work with a friend. Copy the table below into your workbook.
2. Attach a length of tape to the trolley.
3. Run the other end of the tape through the ticker timer.
4. Start the ticker timer going and roll the trolley down the ramp.
5. Repeat steps 1 - 3.
6. On each piece of tape, measure the distance between successive dots. Note these distances in the table below.
7. Use the frequency of the ticker timer to work out the time intervals between successive dots. Note these times in the table below,
8. Work out the average values for distance and time.
9. Use the average distance and average time values to plot a graph of “Distance vs. Time" onto graph paper . Stick the graph paper into your workbook. (Remember that “A vs. B" always means “y vs. x").
10. Insert all axis labels and units onto your graph.
11. Draw the best straight line through your data points.

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
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
characteristics of micro business
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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