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

As for one-dimensional kinematics, we use arrows to represent vectors. The length of the arrow is proportional to the vector’s magnitude. The arrow’s length is indicated by hash marks in [link] and [link] . The arrow points in the same direction as the vector. For two-dimensional motion, the path of an object can be represented with three vectors: one vector shows the straight-line path between the initial and final points of the motion, one vector shows the horizontal component of the motion, and one vector shows the vertical component of the motion. The horizontal and vertical components of the motion add together to give the straight-line path. For example, observe the three vectors in [link] . The first represents a 9-block displacement east. The second represents a 5-block displacement north. These vectors are added to give the third vector, with a 10.3-block total displacement. The third vector is the straight-line path between the two points. Note that in this example, the vectors that we are adding are perpendicular to each other and thus form a right triangle. This means that we can use the Pythagorean theorem to calculate the magnitude of the total displacement. (Note that we cannot use the Pythagorean theorem to add vectors that are not perpendicular. We will develop techniques for adding vectors having any direction, not just those perpendicular to one another, in Vector Addition and Subtraction: Graphical Methods and Vector Addition and Subtraction: Analytical Methods .)

Questions & Answers

tree physical properties of heat
Bello Reply
tree is a type of organism that grows very tall and have a wood trunk and branches with leaves... how is that related to heat? what did you smoke man?
what are the uses of dimensional analysis
Racheal Reply
Dimensional Analysis. The study of relationships between physical quantities with the help of their dimensions and units of measurements is called dimensional analysis. We use dimensional analysis in order to convert a unit from one form to another.
Emmanuel
meaning of OE and making of the subscript nc
ferunmi Reply
can I ask a question
Negash
kinetic functional force
Moyagabo Reply
what is a principal wave?
Haider Reply
A wave the movement of particles on rest position transferring energy from one place to another
Gabche
not wave. i need to know principal wave or waves.
Haider
principle wave is a superposition of wave when two or more waves meet at a point , whose amplitude is the algebraic sum of the amplitude of the waves
arshad
kindly define principal wave not principle wave (principle of super position) if u can understand my question
Haider
what is a model?
Ella Reply
hi
Muhanned
why are electros emitted only when the frequency of the incident radiation is greater than a certain value
ANSELEM Reply
b/c u have to know that for emission of electron need specific amount of energy which are gain by electron for emission . if incident rays have that amount of energy electron can be emitted, otherwise no way.
Nazir
search photoelectric effect on Google
Nazir
what is ohm's law
Pamilerin Reply
states that electric current in a given metallic conductor is directly proportional to the potential difference applied between its end, provided that the temperature of the conductor and other physical factors such as length and cross-sectional area remains constant. mathematically V=IR
ANIEFIOK
hi
Gundala
A body travelling at a velocity of 30ms^-1 in a straight line is brought to rest by application of brakes. if it covers a distance of 100m during this period, find the retardation.
Pamilerin Reply
just use v^2-u^2=2as
Gundala
how often does electrolyte emits?
alhassan
just use +€^3.7°√π%-4¢•∆¥%
v^2-u^2=2as v=0,u=30,s=100 -30^2=2a*100 -900=200a a=-900/200 a=-4.5m/s^2
akinyemi
what is distribution of trade
Grace Reply
what's acceleration
Joshua Reply
The change in position of an object with respect to time
Mfizi
Acceleration is velocity all over time
Pamilerin
hi
Stephen
It's not It's the change of velocity relative to time
Laura
Velocity is the change of position relative to time
Laura
acceleration it is the rate of change in velocity with time
Stephen
acceleration is change in velocity per rate of time
Noara
what is ohm's law
Stephen
Ohm's law is related to resistance by which volatge is the multiplication of current and resistance ( U=RI)
Laura
acceleration is the rate of change. of displacement with time.
Radical
the rate of change of velocity is called acceleration
Asma
how i don understand
Willam Reply
how do I access the Multiple Choice Questions? the button never works and the essay one doesn't either
Savannah Reply
How do you determine the magnitude of force
Peace Reply
mass × acceleration OR Work done ÷ distance
Seema
Which eye defect is corrected by a lens having different curvatures in two perpendicular directions?
Valentina Reply
acute astigmatism?
Practice Key Terms 1

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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