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Learning objectives

By the end of this section, you will be able to:

  • Observe that motion in two dimensions consists of horizontal and vertical components.
  • Understand the independence of horizontal and vertical vectors in two-dimensional motion.

The information presented in this section supports the following AP® learning objectives and science practices:

  • 3.A.1.1 The student is able to express the motion of an object using narrative, mathematical, and graphical representations. (S.P. 1.5, 2.1, 2.2)
  • 3.A.1.2 The student is able to design an experimental investigation of the motion of an object. (S.P. 4.2)
  • 3.A.1.3 The student is able to analyze experimental data describing the motion of an object and is able to express the results of the analysis using narrative, mathematical, and graphical representations. (S.P. 5.1)
A busy traffic intersection in New York showing vehicles moving on the road.
Walkers and drivers in a city like New York are rarely able to travel in straight lines to reach their destinations. Instead, they must follow roads and sidewalks, making two-dimensional, zigzagged paths. (credit: Margaret W. Carruthers)

Two-dimensional motion: walking in a city

Suppose you want to walk from one point to another in a city with uniform square blocks, as pictured in [link] .

An X Y graph with origin at zero zero with x axis labeled nine blocks east and y axis labeled five blocks north. Starting point at the origin and destination at point nine on the x axis and point five on the y axis.
A pedestrian walks a two-dimensional path between two points in a city. In this scene, all blocks are square and are the same size.

The straight-line path that a helicopter might fly is blocked to you as a pedestrian, and so you are forced to take a two-dimensional path, such as the one shown. You walk 14 blocks in all, 9 east followed by 5 north. What is the straight-line distance?

An old adage states that the shortest distance between two points is a straight line. The two legs of the trip and the straight-line path form a right triangle, and so the Pythagorean theorem, a 2  +  b 2  =  c 2 size 12{a rSup { size 8{2} } " + "b rSup { size 8{2} } " = "c rSup { size 8{2} } } {} , can be used to find the straight-line distance.

A right-angled triangle with base labeled a height labeled b and hypotenuse labeled c is shown. Using Pythagorean theorem c is calculated as square root of a squared plus b squared.
The Pythagorean theorem relates the length of the legs of a right triangle, labeled a size 12{a} {} and b size 12{b} {} , with the hypotenuse, labeled c size 12{c} {} . The relationship is given by: a 2 b 2 c 2 size 12{a rSup { size 8{2} }  "+ "b rSup { size 8{2} }  "= "c rSup { size 8{2} } } {} . This can be rewritten, solving for c size 12{A} {} : c  =  a 2 b 2 size 12{c" = " sqrt {a rSup { size 8{2} }  "+ "b rSup { size 8{2} } } } {} .

The hypotenuse of the triangle is the straight-line path, and so in this case its length in units of city blocks is ( 9 blocks ) 2 ( 5 blocks ) 2 = 10 . 3 blocks size 12{ sqrt { \( "9 blocks" \) rSup { size 8{2} }  "+ " \( "5 blocks" \) rSup { size 8{2} } }  "= 10" "." "3 blocks"} {} , considerably shorter than the 14 blocks you walked. (Note that we are using three significant figures in the answer. Although it appears that “9” and “5” have only one significant digit, they are discrete numbers. In this case “9 blocks” is the same as “9.0 or 9.00 blocks.” We have decided to use three significant figures in the answer in order to show the result more precisely.)

An X Y graph with origin at zero zero with x-axis labeled nine blocks east and y axis labeled five blocks north. A diagonal vector arrow joining starting point at point zero on x axis and destination at point five on y axis with its direction northeast is shown. A helicopter is flying along the diagonal vector arrow with helicopter path of ten point three blocks. The angle formed by diagonal vector arrow and the x-axis is equal to twenty-nine point one degrees.
The straight-line path followed by a helicopter between the two points is shorter than the 14 blocks walked by the pedestrian. All blocks are square and the same size.

The fact that the straight-line distance (10.3 blocks) in [link] is less than the total distance walked (14 blocks) is one example of a general characteristic of vectors. (Recall that vectors are quantities that have both magnitude and direction.)

Questions & Answers

what are the basic statistical concepts in psychological research ?
Pawanjeet Reply
mean ,median ,NPC ,t ,z test
Soumyashree
what is the NPC?
Fouad
Maen median mood stander daviation
Janat
Percentile also
Janat
NPC stans for normal probability curve
Janat
Normal probability curve (NPC)
Janat
yez
Soumyashree
Yhn koi b questione kr skta h?
Janat
is there any scale for measuring post COVID Anxiety?
Richard Reply
I'd like to know if there's any?
Zino
I've heard about Coronavirus Anxiety Scale (CAS). It works as a mental health screener but I'm not sure anyone can do the test except psychologist
divani
Thanks !
Zino
no
Hausa
Thanks Divani zahra
Mohd
explain the function how intelligence and learning function together
Truvanie Reply
Learning comes before intelligence Learning could be the process of acquiring knowledge. Or the ability to gain knowledge Whereas intelligence is the display of acquired knowledge.
Felix
schools of psychology
Onogwu Reply
hi
giri
hi
Swapnil
hi
Munir
hello
Imsekar
hai
Tu
Hi
piya
hello
kajal
Hello
Mohamed
hii
grace
hi
george
hi
Anuja
hi
kajal
gud night
grace
Good night everyone
Anuja
good night 😴🌃
Swapnil
hello
Ananya
What's the point of this chat section?
Mohamed
to say hello and to feel solidarity.
Tu
the early childhood can reference the emotions?
Sammi Reply
Our early childhood experiences are important in the development of our personalities. It kind of shapes our behaviour and mannerism.
Debarpita
Especially in choosing career
Felix
😅Well that's complicated. It differs from person to person.
Debarpita
The early childhood or let's say your upbringing plays a vital role in driving your emotions. If one did not have a pleasant childhood it's highly likely they are more vulnerable to depression or some other psychological condition.
Indefatigable
how to change the emotion?
Sammi Reply
change the thought at hand .
Shawnte
Then I'm assuming the mind will begin to let go of the prior emotion.
Shawnte
Yes right. Think about something else, then emotion will follow.
Annie
then join life
Munir
I don't understand about this
Munir
Emotions run hand in hand with the mind and environment. A person can be sad or angry on a certain things going on in their mind then something can happen in the environment around them that m adds keys them happy.
Douglas
how has the new legalization of marijuana effected the anti drug programs
Summer Reply
how to change the emotional ?
Sammi
I am wrilly interisting about big problems psihycil nature between famillys/like parents bully some childe physically , even hall life and show some kinde of heath and $inaid
Zorica Reply
same
Gaea
Never give minor issues space. Time creates space, more time more space therefore if u give a small issue more time it gets more space in your mind hence breeding stress
Andrew Reply
yes yes yes so true
Gaea
hi..my brother gets angry Everytime I say no to him for anything he ask for he don't talk to me after that.. Everytime I only have to say sorry so this time I decided not to say sorry he haven't talked to me for two weeks I am so stressed what should I do
Nush
give him so time. I think you should let him figure out his own feelings. plus, you did the right thing. he needs to know that not everyone will give him what he wants.
Gaea
Don't be stressed out . You did right thing. but remember one thing don't show angry gestures to him or don't try to make him realize by teasing or scolding. If you try to be humble and patient, he may understand his mistake on his own.
PARUL
yeah
Gaea
Hi everyone! So i'm gonna be a college freshman soon and i'm gonna major in Psychology😬 but lately i've been worrying about what job i'm gonna have after getting my Bachelor's degree. Any tips or advice?
Angelique Reply
I think minimum of a masters degree is needed in order to get a job once you do mphil and phd u gonna get licence to work as a clinical psychologist and a lot more options 😊best wishes
kitty
hi...my brother gets angry with me anytime I say no to him for anything he ask for and stop talking to me every time I only have to say sorry..so I decided this time not to say sorry he haven't talked to me at all what should I do please help me
Nush
I think it's just a matter of ego and pride.they may have a kind of feeling like if they apologize it hurts their integrity and pride. Some people never admit they are wrong and it creates stress in the other person. .some people are not empathetic and they always want to feel cared n lovedbyother
kitty
so what can we do..if you are a person who always says sorry it's a good indicator that you have such a broad mind where you are ready to solve problems n you love them even though they are bad at you..basically everyone in this whole world is selfish..humans are selfish but there are some people
kitty
like you whom God created just to solve problems and lead other people in a better way..what you should do is create a beautiful bonding bw your bro ask him softly about his problems..love him more..and eventually he may become softer and lovable
kitty
if your bro is an introvert there could be chances where he has got several problems he haven't discussed with anybody which brings out their egoistic behr and blunt nature...ask him to say about his problems..make him understand the importance of apologies and empathetic behr..and try not to end up
kitty
in a fight discussing these things...u should softly deal with it..even if other people let you down or didn't say sorry to you doesn't mean that you r wrong..u are saying sorry everytime because u have a good heart..
kitty
hope this helps🐒have a good day..good life!!😊
kitty
hi..I have a problem.i will emotional fast, every time when I will give any seminar or etc I will get so emotional and I will cry on the stage....what can I do? give me a solution please
Secular Reply
try to think why your on stage and the audience arnt, try to stay positive and maybe meditate before your performance
Raneem
well i personally suggest trying to practice a lot to other people or to yourself be4 stuff like seminars (like rehearsing), but i think you should ask other people too. also like quora is a good place to ask questions and u might get more/better feedback
Aon
i know this is a psychology app, but quora is specifically designed for asking and answering questions (also ask friends/family/etc)
Aon
Coz u scared of society so that ur confidence is very low ...
Arishfa
Take deep breaths it might sound silly but it actually works.. calms your body and mind.don not focus on things like it may go wrong...or what people will think...just prepare and present..💖
kitty
Being emotionl is nt a fault,its gud dat u atleast knw abt it.Accept ur feelings/emotions.Dis time know wht's mking u cry.Is it d topic or ur intrnl feeling abt ur performance?if it's abt prfrmnce jus know dat none s perfect & b confident in ur subjct by practicing it.Remember None can play ur role.
sailaja
i had a problem that;even though i have people around me, I'm feeling lonely
sharath
don't be @sharath
Gaea
is there any online facilities to acquire Internship?
Tanvi Reply
psychology is the study of human behaviour,mental processes and experiences.It id because behaviour cant be stable over a period of time hence,it considered subjective.morover, human experiences cant be objective
Rija Reply
Is this a Statement OR a Question ? all things considered, psch has come a long way in its relatively short life. i was getting an "art" degree, when I enrolled, the Pseudo science changed and I landed a science degree instead (worth more when employers look at it)
Lyric
human behavior is fascinating if you ask me, my 1st degree was criminology/investigations, the ultimate goal of the field is control <~~ sounds terrible but in reality it's more about learning to predict the behaviors we need prevented & find a way to stop them before the next school shooting or btk
Lyric
It's needs to be tackle at young age before it heads to the teens stage then sometimes they don't wanna hear it or un cool
Derek
the suffix ology is study of
Mouze Reply
the prefix psyche is the soul
Mouze
iwillbecomeasuideideaseveryday whatareiwilldo
Vijaya Reply
?
BLYTHE
@Vjaya are you OK?
Ashantelle
u should have hope in life...its not that hard as you think...dont give up even though I don't know what you are talking about just pray..breathe...relax..eveything will be fine 💖
kitty
would you like to talk? I'm an advocate for suicide prevention .
Sonya
that's quite gud
sharath
listen buddy if you keep having those thoughts you will manifest bad energy, i recommend listening to positive subliminals for your subconscious mind, and try being positive, it doesn't have to be real just fake it until you make it
stan
ohhh a Loona here?
here a Orbit too
Vijaya I wish I could Help you-
if you're having suicidal thoughts or anything like that, please contact a suicide hotline b/c i think they could help u more more than we could 1-800-273-8255 there are probs other ones just search it
Aon
how are you feeling tho? im kinda confused by the question and i didnt even know this app had a discussion thing
Aon
please call suicide hotline though if you are having suucidal thoughts because they can really help you (definitely more than some of us could) 1-800-273-8255
Aon
please call suicide hotline though if you are having suucidal thoughts because they can really help you (definitely more than some of us could)
Aon
is anyone there
Robin
@vijaya if ur still reading comments, idrk what u meant but please do stay safe and get help if you're feeling badly
Aon
Hii🙋🏽‍♀️
Angelique
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Source:  OpenStax, Sample chapters: openstax college physics for ap® courses. OpenStax CNX. Oct 23, 2015 Download for free at http://legacy.cnx.org/content/col11896/1.9
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