<< Chapter < Page Chapter >> Page >
a = Δ v t

where Δ v is the change in velocity, i.e. Δ v = v f - v i . Thus we have

a = v f - v i t v f = v i + a t

Derivation of [link]

We have seen that displacement can be calculated from the area under a velocity vs. time graph. For uniformly accelerated motion the most complicated velocity vs. time graph we can have is a straight line. Look at the graph below - it represents an object with a starting velocity of v i , accelerating to a final velocity v f over a total time t .

To calculate the final displacement we must calculate the area under the graph - this is just the area of the rectangle added to the area of the triangle. This portion of the graph has been shaded for clarity.

Area = 1 2 b × h = 1 2 t × ( v f - v i ) = 1 2 v f t - 1 2 v i t
Area = × b = t × v i = v i t
Displacement = Area + Area Δ x = v i t + 1 2 v f t - 1 2 v i t Δ x = ( v i + v f ) 2 t

Derivation of [link]

This equation is simply derived by eliminating the final velocity v f in [link] . Remembering from [link] that

v f = v i + a t

then [link] becomes

Δ x = v i + v i + a t 2 t = 2 v i t + a t 2 2 Δ x = v i t + 1 2 a t 2

Derivation of [link]

This equation is just derived by eliminating the time variable in the above equation. From [link] we know

t = v f - v i a

Substituting this into [link] gives

Δ x = v i ( v f - v i a ) + 1 2 a ( v f - v i a ) 2 = v i v f a - v i 2 a + 1 2 a ( v f 2 - 2 v i v f + v i 2 a 2 ) = v i v f a - v i 2 a + v f 2 2 a - v i v f a + v i 2 2 a 2 a Δ x = - 2 v i 2 + v f 2 + v i 2 v f 2 = v i 2 + 2 a Δ x

This gives us the final velocity in terms of the initial velocity, acceleration and displacement and is independent of the time variable.

A racing car is travelling north. It accelerates uniformly covering a distance of 725 m in 10 s. If it has an initial velocity of 10 m · s - 1 , find its acceleration.

  1. We are given:

    v i = 10 m · s - 1 Δ x = 725 m t = 10 s a = ?
  2. If you struggle to find the correct equation, find the quantity that is not given and then look for an equation that has this quantity in it.

    We can use equation [link]

    Δ x = v i t + 1 2 a t 2
  3. Δ x = v i t + 1 2 a t 2 725 m = ( 10 m · s - 1 × 10 s ) + 1 2 a × ( 10 s ) 2 725 m - 100 m = ( 50 s 2 ) a a = 12 , 5 m · s - 2
  4. The racing car is accelerating at 12,5 m · s - 2 north.

A motorcycle, travelling east, starts from rest, moves in a straight line with a constant acceleration and covers a distance of 64 m in 4 s. Calculate

  1. its acceleration
  2. its final velocity
  3. at what time the motorcycle had covered half the total distance
  4. what distance the motorcycle had covered in half the total time.
  1. We are given:

    v i = 0 m · s - 1 ( because the object starts from rest. ) Δ x = 64 m t = 4 s a = ? v f = ? t = ? at half the distance Δ x = 32 m . Δ x = ? at half the time t = 2 s .

    All quantities are in SI units.

  2. We can use [link]

    Δ x = v i t + 1 2 a t 2
  3. Δ x = v i t + 1 2 a t 2 64 m = ( 0 m · s - 1 × 4 s ) + 1 2 a × ( 4 s ) 2 64 m = ( 8 s 2 ) a a = 8 m · s - 2 east
  4. We can use [link] - remember we now also know the acceleration of the object.

    v f = v i + a t
  5. v f = v i + a t v f = 0 m · s - 1 + ( 8 m · s - 2 ) ( 4 s ) = 32 m · s - 1 east
  6. We can use [link] :

    Δ x = v i + 1 2 a t 2 32 m = ( 0 m · s - 1 ) t + 1 2 ( 8 m · s - 2 ) ( t ) 2 32 m = 0 + ( 4 m · s - 2 ) t 2 8 s 2 = t 2 t = 2 , 83 s
  7. Half the time is 2 s, thus we have v i , a and t - all in the correct units. We can use [link] to get the distance:

    Δ x = v i t + 1 2 a t 2 = ( 0 ) ( 2 ) + 1 2 ( 8 ) ( 2 ) 2 = 16 m east
    1. The acceleration is 8 m · s - 2 east
    2. The velocity is 32 m · s - 1 east
    3. The time at half the distance is 2,83 s
    4. The distance at half the time is 16 m east

Questions & Answers

what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
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?
s. Reply
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
Devang Reply
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
Abhijith Reply
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?
s. Reply
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 ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Physics - grade 10 [caps 2011]. OpenStax CNX. Jun 14, 2011 Download for free at http://cnx.org/content/col11298/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Physics - grade 10 [caps 2011]' conversation and receive update notifications?

Ask