What Is the Displacement if Your Travels 5 Km North Then 3 Km East and Then 1 Km North Again

three Motion Along a Straight Line

three.1 Position, Displacement, and Average Velocity

Learning Objectives

Past the end of this department, you will be able to:

Define position, displacement, and distance traveled.
Summate the full displacement given the position as a role of time.
Make up one's mind the total distance traveled.
Calculate the average velocity given the displacement and elapsed time.

When you're in motion, the basic questions to ask are: Where are you? Where are you going? How fast are yous getting there? The answers to these questions crave that you specify your position, your displacement, and your average velocity—the terms we define in this section.

Position

To describe the motility of an object, yous must showtime exist able to describe its position (10): where it is at any item time. More precisely, we need to specify its position relative to a user-friendly frame of reference. A frame of reference is an arbitrary gear up of axes from which the position and motion of an object are described. Globe is often used as a frame of reference, and we often describe the position of an object as it relates to stationary objects on Earth. For example, a rocket launch could be described in terms of the position of the rocket with respect to Earth as a whole, whereas a cyclist'south position could be described in terms of where she is in relation to the buildings she passes (Figure). In other cases, nosotros employ reference frames that are not stationary but are in movement relative to Earth. To draw the position of a person in an airplane, for case, we apply the airplane, not Earth, as the reference frame. To describe the position of an object undergoing one-dimensional motion, we often employ the variable x. Later in the chapter, during the word of free fall, we employ the variable y.

Picture shows three people riding bicycles next to a canal. — **Effigy 3.ii** These cyclists in Vietnam can exist described past their position relative to buildings or a canal. Their movement tin can be described by their change in position, or displacement, in a frame of reference. (credit: Suzan Blackness)

Displacement

If an object moves relative to a frame of reference—for instance, if a professor moves to the right relative to a whiteboard (Figure)—and then the object's position changes. This alter in position is called deportation. The word displacement implies that an object has moved, or has been displaced. Although position is the numerical value of ten along a straight line where an object might exist located, displacement gives the change in position along this line. Since displacement indicates direction, information technology is a vector and can be either positive or negative, depending on the choice of positive direction. Also, an assay of motility can have many displacements embedded in information technology. If right is positive and an object moves 2 m to the right, so 4 thou to the left, the individual displacements are 2 g and

$-4$

chiliad, respectively.

Illustration shows professor at two different locations. The first location is marked as 1.5 meters at the x axis; the second location is marked as 3.5 meters at the x axis. The displacement between the two locations is 2 meters. — **Figure 3.3** A professor paces left and correct while lecturing. Her position relative to Earth is given by ten. The +two.0-one thousand displacement of the professor relative to Earth is represented by an arrow pointing to the correct.

Displacement

Displacement

$\text{Δ}x$

is the change in position of an object:

$\text{Δ}x={x}_{\text{f}}-{x}_{0},$

where

$\text{Δ}x$

is displacement,

${x}_{\text{f}}$

is the final position, and

${x}_{0}$

is the initial position.

We employ the majuscule Greek letter delta (Δ) to mean "change in" whatever quantity follows it; thus,

$\text{Δ}x$

means modify in position (final position less initial position). We ever solve for displacement past subtracting initial position

${x}_{0}$

from last position

${x}_{\text{f}}$

. Annotation that the SI unit for displacement is the meter, merely sometimes nosotros use kilometers or other units of length. Keep in mind that when units other than meters are used in a trouble, you may demand to convert them to meters to complete the calculation (meet Conversion Factors).

Objects in move can also have a serial of displacements. In the previous example of the pacing professor, the private displacements are two m and

$-4$

m, giving a total displacement of −2 m. Nosotros define total deportation

$\text{Δ}{x}_{\text{Total}}$

, as the sum of the individual displacements, and express this mathematically with the equation

$\text{Δ}{x}_{\text{Total}}=\sum \text{Δ}{x}_{\text{i}},$

where

$\text{Δ}{x}_{i}$

are the individual displacements. In the earlier example,

$\text{Δ}{x}_{1}={x}_{1}-{x}_{0}=2-0=2\,\text{m.}$

Similarly,

$\text{Δ}{x}_{2}={x}_{2}-{x}_{1}=-2-(2)=-4\,\text{m.}$

Thus,

$\text{Δ}{x}_{\text{Total}}=\text{Δ}{x}_{1}+\text{Δ}{x}_{2}=2-4=-2\,\text{m}\text{.}$

The total displacement is 2 − 4 = −2 1000 to the left, or in the negative management. Information technology is also useful to calculate the magnitude of the displacement, or its size. The magnitude of the deportation is always positive. This is the absolute value of the displacement, considering displacement is a vector and cannot have a negative value of magnitude. In our example, the magnitude of the total displacement is 2 yard, whereas the magnitudes of the private displacements are 2 g and 4 m.

The magnitude of the total displacement should non exist confused with the distance traveled. Altitude traveled

${x}_{\text{Total}}$

, is the total length of the path traveled betwixt ii positions. In the previous problem, the distance traveled is the sum of the magnitudes of the individual displacements:

${x}_{\text{Total}}=|\text{Δ}{x}_{1}|+|\text{Δ}{x}_{2}|=2+4=6\,\text{m}\text{.}$

Average Velocity

To calculate the other physical quantities in kinematics we must introduce the time variable. The time variable allows us not only to state where the object is (its position) during its motion, but also how fast it is moving. How fast an object is moving is given by the rate at which the position changes with time.

For each position

${x}_{\text{i}}$

, nosotros assign a particular time

${t}_{\text{i}}$

. If the details of the motion at each instant are non important, the rate is usually expressed equally the average velocity

$\overset{\text{-}}{v}$

. This vector quantity is but the total displacement between two points divided by the time taken to travel between them. The time taken to travel between two points is called the elapsed time

$\text{Δ}t$

Average Velocity

${x}_{1}$

and

${x}_{2}$

are the positions of an object at times

${t}_{1}$

and

${t}_{2}$

, respectively, then

$\begin{array}{cc} \text{Average velocity}=\overset{\text{-}}{v}=\frac{\text{Displacement between two points}}{\text{Elapsed time between two points}}\\ \overset{\text{-}}{v}=\frac{\text{Δ}x}{\text{Δ}t}=\frac{{x}_{2}-{x}_{1}}{{t}_{2}-{t}_{1}}.\end{array}$

It is important to note that the average velocity is a vector and can exist negative, depending on positions

${x}_{1}$

and

${x}_{2}$

Example

Delivering Flyers

Jill sets out from her domicile to deliver flyers for her one thousand auction, traveling due east along her street lined with houses. At

$0.5$

km and ix minutes later she runs out of flyers and has to retrace her steps back to her business firm to get more than. This takes an boosted 9 minutes. After picking up more flyers, she sets out again on the same path, standing where she left off, and ends up 1.0 km from her house. This third leg of her trip takes

$15$

minutes. At this point she turns back toward her business firm, heading west. After

$1.75$

km and

$25$

minutes she stops to rest.

What is Jill'south total displacement to the bespeak where she stops to rest?
What is the magnitude of the final displacement?
What is the average velocity during her entire trip?
What is the total distance traveled?
Make a graph of position versus fourth dimension.

A sketch of Jill's movements is shown in (Effigy).

Figure shows a timeline of a person's movement. First displacement is from the home to the right by 0.5 kilometers. Second displacement is back to the starting point. Third displacement is to the right by 1.0 kilometer. Fourth displacement is from the final point to the left by 1.75 kilometers. — Figure 3.4 Timeline of Jill's movements.

Strategy

The problem contains information on the diverse legs of Jill'south trip, so it would be useful to make a table of the physical quantities. We are given position and time in the diction of the trouble so nosotros can calculate the displacements and the elapsed fourth dimension. Nosotros take east to be the positive direction. From this information nosotros can observe the total deportation and boilerplate velocity. Jill's dwelling house is the starting signal

${x}_{0}$

. The following tabular array gives Jill's time and position in the first two columns, and the displacements are calculated in the third cavalcade.

Time t _i (min)	Position ${x}_{i}$ (km)	Deportation $\text{Δ}{x}_{\text{i}}$ (km)
${t}_{0}=0$	${x}_{0}=0$	$\text{Δ}{x}_{0}=0$
${t}_{1}=9$	${x}_{1}=0.5$	$\text{Δ}{x}_{1}={x}_{1}-{x}_{0}=0.5$
${t}_{2}=18$	${x}_{2}=0$	$\text{Δ}{x}_{2}={x}_{2}-{x}_{1}=-0.5$
${t}_{3}=33$	${x}_{3}=1.0$	$\text{Δ}{x}_{3}={x}_{3}-{x}_{2}=1.0$
${t}_{4}=58$	${x}_{4}=-0.75$	$\text{Δ}{x}_{4}={x}_{4}-{x}_{3}=-1.75$

Solution

[reveal-answer q="905360″]Show Reply[/reveal-answer]
[hidden-reply a="905360″]From the to a higher place tabular array, the full deportation is
$\sum \text{Δ}{x}_{\text{i}}=0.5-0.5+1.0-1.75\,\text{km}=-0.75\,\text{km}\text{.}$

[/hidden-reply]
[reveal-answer q="289407″]Show Answer[/reveal-answer]
[subconscious-answer a="289407″]The magnitude of the total deportation is
$|-0.75|\,\text{km}=0.75\,\text{km}$

.[/hidden-answer]
[reveal-respond q="368664″]Prove Answer[/reveal-respond]
[hidden-respond a="368664″]
$\text{Average velocity}=\frac{\text{Total}\,\text{displacement}}{\text{Elapsed}\,\text{time}}=\overset{\text{-}}{v}=\frac{-0.75\,\text{km}}{58\,\text{min}}=-0.013\,\text{km/min}$

[/subconscious-answer]
[reveal-answer q="427772″]Show Answer[/reveal-respond]
[hidden-respond a="427772″]The full altitude traveled (sum of magnitudes of individual displacements) is
${x}_{\text{Total}}=\sum |\text{Δ}{x}_{\text{i}}|=0.5+0.5+1.0+1.75\,\text{km}=3.75\,\text{km}$

.[/subconscious-answer]
[reveal-reply q="329442″]Show Answer[/reveal-answer]
[subconscious-answer a="329442″]We can graph Jill's position versus fourth dimension as a useful assistance to see the motion; the graph is shown in (Effigy).

Figure iii.v This graph depicts Jill'south position versus time. The average velocity is the gradient of a line connecting the initial and final points.

[/hidden-answer]

Significance

Jill'south total displacement is −0.75 km, which means at the stop of her trip she ends up

$0.75\,\text{km}$

due w of her home. The boilerplate velocity means if someone was to walk due w at

$0.013$

km/min starting at the same time Jill left her home, they both would make it at the final stopping point at the aforementioned fourth dimension. Note that if Jill were to stop her trip at her house, her full displacement would exist cipher, too as her average velocity. The total altitude traveled during the 58 minutes of elapsed time for her trip is iii.75 km.

Check Your Understanding

A cyclist rides three km westward so turns around and rides two km east. (a) What is his deportation? (b) What is the distance traveled? (c) What is the magnitude of his displacement?

Figure shows timeline of cyclist's movement. First displacement is to the left by 3.0 kilometers. Second displacement is from the final point to the right by 2.0 kilometers.

[reveal-answer q="151192″]Bear witness Reply[/reveal-answer]
[subconscious-reply a="151192″](a) The rider'south displacement is

$\text{Δ}x={x}_{\text{f}}-{x}_{0}=-1\,\text{km}$

. (The displacement is negative because we accept east to exist positive and west to be negative.) (b) The distance traveled is 3 km + 2 km = five km. (c) The magnitude of the deportation is 1 km.[/hidden-answer]

Summary

Kinematics is the description of move without considering its causes. In this chapter, information technology is express to motion along a straight line, called one-dimensional motion.
Displacement is the change in position of an object. The SI unit for displacement is the meter. Displacement has direction also as magnitude.
Altitude traveled is the total length of the path traveled between 2 positions.
Time is measured in terms of change. The time betwixt ii position points
${x}_{1}$

and

${x}_{2}$

is

$\text{Δ}t={t}_{2}-{t}_{1}$

. Elapsed time for an event is

$\text{Δ}t={t}_{\text{f}}-{t}_{0}$

, where

${t}_{\text{f}}$

is the final time and

${t}_{0}$

is the initial time. The initial time is often taken to be nothing.
Average velocity
$\overset{\text{-}}{v}$

is divers as deportation divided past elapsed fourth dimension. If

${x}_{1},{t}_{1}$

and

${x}_{2},{t}_{2}$

are two position time points, the average velocity between these points is

$\overset{\text{-}}{v}=\frac{\text{Δ}x}{\text{Δ}t}=\frac{{x}_{2}-{x}_{1}}{{t}_{2}-{t}_{1}}.$

Conceptual Questions

Give an example in which at that place are articulate distinctions amidst distance traveled, displacement, and magnitude of displacement. Identify each quantity in your example specifically.

[reveal-answer q="fs-id1168329491939″]Show Solution[/reveal-answer]

[hidden-answer a="fs-id1168329491939″]

You drive your car into boondocks and render to drive past your firm to a friend's business firm.

[/hidden-answer]

Under what circumstances does distance traveled equal magnitude of deportation? What is the but case in which magnitude of deportation and displacement are exactly the aforementioned?

Leaner movement dorsum and forth using their flagella (structures that look like little tails). Speeds of upwardly to 50 μm/s (50 × 10⁻⁶ g/due south) have been observed. The total distance traveled past a bacterium is large for its size, whereas its displacement is small. Why is this?

[reveal-answer q="fs-id1168326792665″]Show Solution[/reveal-answer]

[subconscious-respond a="fs-id1168326792665″]

If the bacteria are moving dorsum and forth, so the displacements are canceling each other and the concluding displacement is minor.

[/hidden-answer]

Give an example of a device used to measure out time and identify what change in that device indicates a change in fourth dimension.

Does a car's odometer measure altitude traveled or displacement?

[reveal-reply q="fs-id1168329476976″]Evidence Solution[/reveal-answer]

[hidden-answer a="fs-id1168329476976″]

Distance traveled

[/hidden-answer]

During a given time interval the average velocity of an object is cypher. What can you say conclude about its displacement over the time interval?

Problems

Consider a coordinate system in which the positive x axis is directed up vertically. What are the positions of a particle (a) v.0 m directly above the origin and (b) 2.0 one thousand below the origin?

A car is two.0 km west of a traffic light at t = 0 and 5.0 km eastward of the light at t = vi.0 min. Presume the origin of the coordinate system is the lite and the positive ten direction is due east. (a) What are the machine'southward position vectors at these 2 times? (b) What is the car's displacement betwixt 0 min and six.0 min?

[reveal-respond q="fs-id1168329462686″]Show Solution[/reveal-answer]

[hidden-respond a="fs-id1168329462686″]

${\overset{\to }{x}}_{1}=(-2.0\,\text{m})\hat{i}$

${\overset{\to }{x}}_{2}=(5.0\,\text{m})\hat{i}$

; b. vii.0 yard east
[/hidden-reply]

The Shanghai maglev train connects Longyang Road to Pudong International Airport, a distance of thirty km. The journey takes eight minutes on boilerplate. What is the maglev train's average velocity?

The position of a particle moving along the 10-centrality is given past

$x(t)=4.0-2.0t$

k. (a) At what time does the particle cross the origin? (b) What is the deportation of the particle between

$\text{t}=3.0\,\text{s}$

and

$\text{t}=6.0\,\text{s}?$

[reveal-reply q="fs-id1168326770074″]Show Solution[/reveal-respond]

[subconscious-answer a="fs-id1168326770074″]

$t=2.0$

s; b.

$x(6.0)-x(3.0)=-8.0-(-2.0)=-6.0\,\text{m}$

[/hidden-respond]

A cyclist rides viii.0 km eastward for 20 minutes, so he turns and heads west for 8 minutes and three.two km. Finally, he rides east for 16 km, which takes 40 minutes. (a) What is the concluding displacement of the cyclist? (b) What is his average velocity?

On February 15, 2013, a superbolide meteor (brighter than the Sun) entered Earth's atmosphere over Chelyabinsk, Russia, and exploded at an altitude of 23.5 km. Eyewitnesses could feel the intense oestrus from the fireball, and the blast moving ridge from the explosion blew out windows in buildings. The boom wave took approximately two minutes 30 seconds to reach ground level. (a) What was the average velocity of the blast wave? b) Compare this with the speed of audio, which is 343 grand/s at bounding main level.

[reveal-respond q="fs-id1168329517552″]Show Solution[/reveal-answer]

[hidden-respond a="fs-id1168329517552″]

a. 150.0 south,

$\overset{\text{-}}{v}=156.7\,\text{m/s}$

; b. 45.seven% the speed of sound at body of water level
[/hidden-answer]

Glossary

average velocity: the displacement divided by the time over which displacement occurs

deportation: the change in position of an object

distance traveled: the full length of the path traveled between 2 positions

elapsed time: the difference between the ending fourth dimension and the showtime time

kinematics: the clarification of movement through backdrop such equally position, time, velocity, and dispatch

position: the location of an object at a particular time

total displacement: the sum of individual displacements over a given time catamenia

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Source: https://opentextbc.ca/universityphysicsv1openstax/chapter/1-1-position-displacement-and-average-velocity/

What Is the Displacement if Your Travels 5 Km North Then 3 Km East and Then 1 Km North Again

three.1 Position, Displacement, and Average Velocity

Learning Objectives

Position

Displacement

Average Velocity

Example

Delivering Flyers

Strategy

Solution

Significance

Check Your Understanding

Summary

Conceptual Questions

Problems

Glossary

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