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Resultant vectors

  1. Harold walks to school by walking 600 m Northeast and then 500 m N 40 W. Determine his resultant displacement by using accurate scale drawings.
  2. A dove flies from her nest, looking for food for her chick. She flies at a velocity of 2 m · s - 1 on a bearing of 135 and then at a velocity of 1,2 m · s - 1 on a bearing of 230 . Calculate her resultant velocity by using accurate scale drawings.
  3. A squash ball is dropped to the floor with an initial velocity of 2,5 m · s - 1 . It rebounds (comes back up) with a velocity of 0,5 m · s - 1 .
    1. What is the change in velocity of the squash ball?
    2. What is the resultant velocity of the squash ball?

Remember that the technique of addition and subtraction just discussed can only be applied to vectors acting along a straight line. When vectors are not in a straight line, i.e. at an angle to each other, the following method can be used:

A more general algebraic technique

Simple geometric and trigonometric techniques can be used to find resultant vectors.

A man walks 40 m East, then 30 m North. Calculate the man's resultant displacement.

  1. As before, the rough sketch looks as follows:

  2. Note that the triangle formed by his separate displacement vectors and his resultant displacement vector is a right-angle triangle. We can thus use the Theorem of Pythagoras to determine the length of the resultant. Let x R represent the length of the resultant vector. Then:

    x R 2 = ( 40 m ) 2 + ( 30 m ) 2 x R 2 = 2 500 m 2 x R = 50 m
  3. Now we have the length of the resultant displacement vector but not yet its direction. To determine its direction we calculate the angle α between the resultant displacement vector and East, by using simple trigonometry:

    tan α = opposite side adjacent side tan α = 30 40 α = tan - 1 ( 0 , 75 ) α = 36 , 9
  4. The resultant displacement is then 50 m at 36,9 North of East.

    This is exactly the same answer we arrived at after drawing a scale diagram!

In the previous example we were able to use simple trigonometry to calculate the resultant displacement. This was possible since thedirections of motion were perpendicular (north and east). Algebraic techniques, however, are not limited to cases where the vectors to be combined are along the same straight line or at right angles to oneanother. The following example illustrates this.

A man walks from point A to point B which is 12 km away on a bearing of 45 . From point B the man walks a further 8 km east to point C. Calculate the resultant displacement.

  1. B A ^ F = 45 since the man walks initially on a bearing of 45 . Then, A B ^ G = B A ^ F = 45 (parallel lines, alternate angles). Both of these angles are included in the rough sketch.

  2. The resultant is the vector AC. Since we know both the lengths of AB and BC and the included angle A B ^ C , we can use the cosine rule:

    A C 2 = A B 2 + B C 2 - 2 · A B · B C cos ( A B ^ C ) = ( 12 ) 2 + ( 8 ) 2 - 2 · ( 12 ) ( 8 ) cos ( 135 ) = 343 , 8 A C = 18 , 5 km
  3. Next we use the sine rule to determine the angle θ :

    sin θ 8 = sin 135 18 , 5 sin θ = 8 × sin 135 18 , 5 θ = sin - 1 ( 0 , 3058 ) θ = 17 , 8

    To find F A ^ C , we add 45 . Thus, F A ^ C = 62 , 8 .

  4. The resultant displacement is therefore 18,5 km on a bearing of 062,8 .

More resultant vectors

  1. A frog is trying to cross a river. It swims at 3 m · s - 1 in a northerly direction towards the opposite bank. The water is flowing in a westerly direction at 5 m · s - 1 . Find the frog's resultant velocity by using appropriate calculations. Include a rough sketch of the situation in your answer.
  2. Sandra walks to the shop by walking 500 m Northwest and then 400 m N 30 E. Determine her resultant displacement by doing appropriate calculations.

Questions & Answers

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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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
Mukesh Reply
Properties of longitudinal waves
Sharoon Reply

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Source:  OpenStax, Siyavula textbooks: grade 10 physical science [caps]. OpenStax CNX. Sep 30, 2011 Download for free at http://cnx.org/content/col11305/1.7
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