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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

A spherica, concave shaving mirror has a radius of curvature of 32 cm .what is the magnification of a persons face. when it is 12cm to the left of the vertex of the mirror
Alona Reply
did you solve?
Shii
A weather vane is some sort of directional arrow parallel to the ground that may rotate freely in a horizontal plane. A typical weather vane has a large cross-sectional area perpendicular to the direction the arrow is pointing, like a “One Way” street sign. The purpose of the weather vane is to indicate the direction of the wind. As wind blows pa
Kavita Reply
If a prism is fully imersed in water then the ray of light will normally dispersed or their is any difference?
Anurag Reply
the same behavior thru the prism out or in water bud abbot
Ju
If this will experimented with a hollow(vaccum) prism in water then what will be result ?
Anurag
What was the previous far point of a patient who had laser correction that reduced the power of her eye by 7.00 D, producing a normal distant vision power of 50.0 D for her?
Jaydie Reply
What is the far point of a person whose eyes have a relaxed power of 50.5 D?
Jaydie
What is the far point of a person whose eyes have a relaxed power of 50.5 D?
Jaydie
A young woman with normal distant vision has a 10.0% ability to accommodate (that is, increase) the power of her eyes. What is the closest object she can see clearly?
Jaydie
29/20 ? maybes
Ju
In what ways does physics affect the society both positively or negatively
Princewill Reply
how can I read physics...am finding it difficult to understand...pls help
rerry Reply
try to read several books on phy don't just rely one. some authors explain better than other.
Ju
And don't forget to check out YouTube videos on the subject. Videos offer a different visual way to learn easier.
Ju
hope that helps
Ju
I have a exam on 12 february
David Reply
what is velocity
Jiti
the speed of something in a given direction.
Ju
what is a magnitude in physics
Jiti Reply
Propose a force standard different from the example of a stretched spring discussed in the text. Your standard must be capable of producing the same force repeatedly.
Giovani Reply
What is meant by dielectric charge?
It's Reply
what happens to the size of charge if the dielectric is changed?
Brhanu Reply
omega= omega not +alpha t derivation
Provakar Reply
u have to derivate it respected to time ...and as w is the angular velocity uu will relace it with "thita × time""
Abrar
do to be peaceful with any body
Brhanu Reply
the angle subtended at the center of sphere of radius r in steradian is equal to 4 pi how?
Saeed Reply
if for diatonic gas Cv =5R/2 then gamma is equal to 7/5 how?
Saeed
define variable velocity
Ali Reply
Practice Key Terms 1

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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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