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Solution

Because v tot size 12{v rSub { size 8{ bold "tot"} } } {} is the vector sum of the v w and v p , its x - and y -components are the sums of the x - and y -components of the wind and plane velocities. Note that the plane only has vertical component of velocity so v p x = 0 and v p y = v p . That is,

v tot x = v w x size 12{v rSub { size 8{"tot"x} } =v rSub { size 8{wx} } } {}

and

v tot y = v w y + v p . size 12{v rSub { size 8{"tot"y} } =v rSub { size 8{wx} } +v rSub { size 8{p} } "."} {}

We can use the first of these two equations to find v w x size 12{v rSub { size 8{ ital "wx"} } } {} :

v w y = v tot x = v tot cos 110º . size 12{v rSub { size 8{wx} } =v rSub { size 8{"tot"x} } =v rSub { size 8{"tot"} } "cos110" rSup { size 8{o} } "."} {}

Because v tot = 38 . 0 m / s size 12{v rSub { size 8{ ital "tot"} } ="38" "." 0m/s} {} and cos 110º = 0.342 size 12{"cos""110º""=""-""0.342"} {} we have

v w y = ( 38.0 m/s ) ( –0.342 ) = –13 m/s.

The minus sign indicates motion west which is consistent with the diagram.

Now, to find v w y size 12{v rSub { size 8{ ital "wy"} } } {} we note that

v tot y = v w y + v p size 12{v rSub { size 8{"tot"y} } =v rSub { size 8{wx} } +v rSub { size 8{p} } } {}

Here v tot y = v tot sin 110º size 12{v rSub { size 8{"tot"y} } =v rSub { size 8{"tot"} }  = v rSub { size 8{"tot"} }  "sin 110º"} {} ; thus,

v w y = ( 38 . 0 m/s ) ( 0 . 940 ) 45 . 0 m/s = 9 . 29 m/s. size 12{v rSub { size 8{wy} } = \( "38" "." 0" m/s" \) \( 0 "." "940" \) - "45" "." 0" m/s"= - 9 "." "29"" m/s."} {}

This minus sign indicates motion south which is consistent with the diagram.

Now that the perpendicular components of the wind velocity v w x size 12{v rSub { size 8{wx} } } {} and v w y size 12{v rSub { size 8{wy} } } {} are known, we can find the magnitude and direction of v w size 12{v rSub { size 8{w} } } {} . First, the magnitude is

v w = v w x 2 + v w y 2 = ( 13 . 0 m/s ) 2 + ( 9 . 29 m/s ) 2

so that

v w = 16 . 0 m/s . size 12{v rSub { size 8{w} } ="16" "." 0" m/s."} {}

The direction is:

θ = tan 1 ( v w y / v w x ) = tan 1 ( 9 . 29 / 13 . 0 ) size 12{θ="tan" rSup { size 8{ - 1} } \( v rSub { size 8{wy} } /v rSub { size 8{wx} } \) ="tan" rSup { size 8{ - 1} } \( - 9 "." "29"/ - "13" "." 0 \) } {}

giving

θ = 35 . . size 12{θ="35" "." 6º"."} {}

Discussion

The wind’s speed and direction are consistent with the significant effect the wind has on the total velocity of the plane, as seen in [link] . Because the plane is fighting a strong combination of crosswind and head-wind, it ends up with a total velocity significantly less than its velocity relative to the air mass as well as heading in a different direction.

Note that in both of the last two examples, we were able to make the mathematics easier by choosing a coordinate system with one axis parallel to one of the velocities. We will repeatedly find that choosing an appropriate coordinate system makes problem solving easier. For example, in projectile motion we always use a coordinate system with one axis parallel to gravity.

Relative velocities and classical relativity

When adding velocities, we have been careful to specify that the velocity is relative to some reference frame . These velocities are called relative velocities . For example, the velocity of an airplane relative to an air mass is different from its velocity relative to the ground. Both are quite different from the velocity of an airplane relative to its passengers (which should be close to zero). Relative velocities are one aspect of relativity    , which is defined to be the study of how different observers moving relative to each other measure the same phenomenon.

Nearly everyone has heard of relativity and immediately associates it with Albert Einstein (1879–1955), the greatest physicist of the 20th century. Einstein revolutionized our view of nature with his modern theory of relativity, which we shall study in later chapters. The relative velocities in this section are actually aspects of classical relativity, first discussed correctly by Galileo and Isaac Newton. Classical relativity is limited to situations where speeds are less than about 1% of the speed of light—that is, less than 3,000 km/s size 12{"3,000 km/s"} {} . Most things we encounter in daily life move slower than this speed.

Let us consider an example of what two different observers see in a situation analyzed long ago by Galileo. Suppose a sailor at the top of a mast on a moving ship drops his binoculars. Where will it hit the deck? Will it hit at the base of the mast, or will it hit behind the mast because the ship is moving forward? The answer is that if air resistance is negligible, the binoculars will hit at the base of the mast at a point directly below its point of release. Now let us consider what two different observers see when the binoculars drop. One observer is on the ship and the other on shore. The binoculars have no horizontal velocity relative to the observer on the ship, and so he sees them fall straight down the mast. (See [link] .) To the observer on shore, the binoculars and the ship have the same horizontal velocity, so both move the same distance forward while the binoculars are falling. This observer sees the curved path shown in [link] . Although the paths look different to the different observers, each sees the same result—the binoculars hit at the base of the mast and not behind it. To get the correct description, it is crucial to correctly specify the velocities relative to the observer.

Questions & Answers

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faith Reply
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faith
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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
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ferunmi Reply
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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
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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
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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
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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
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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
Practice Key Terms 5

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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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