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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

What is physics?
Jeuloriz Reply
physics is a branch of science in which we are dealing with the knowledge of our physical things. macroscopic as well as microscopic. we are going look inside the univers with the help of physics. you can learn nature with the help of physics. so many branches of physics you have to learn physics.
vijay
What are quarks?
Breanna Reply
6 type of quarks
Neyaz
what is candela
Akani Reply
Candela is the unit for the measurement of light intensity.
Osei
any one can prove that 1hrpower= 746 watt
Neyaz Reply
Newton second is the unit of ...............?
Neyaz
Impulse and momentum
Fauzia
force×time and mass× velocity
vijay
Good
Neyaz
What is the simple harmonic motion?
Fauzia Reply
oscillatory motion under a retarding force proportional to the amount of displacement from an equilibrium position
Yuri
Straight out of google, you could do that to, I suppose.
Yuri
*too
Yuri
ok
Fauzia
Oscillatory motion under a regarding force proportional to the amount of displacement from an equilibrium position
Neyaz
examples of work done by load of gravity
Maureen Reply
What is ehrenfest theorem?
Fauzia Reply
You can look it up, faster and more reliable answer.
Yuri
That isn't a question to ask on a forum and I also have no idea what that is.
Yuri
what is the work done by gravity on the load 87kj,11.684m,mass xkg[g=19m/s
Maureen
What is law of mass action?
Fauzia Reply
rate of chemical reactions is proportional to concentration of reactants ...
muhammad
ok thanks
Fauzia
what is lenses
Ndobe Reply
lenses are two types
Fauzia
concave and convex
muhammad
right
Fauzia
speed of light in space
Vikash Reply
in vacuum speed of light is 3×10^8 m/s
vijay
ok
Vikash
2.99×10^8m/s
Umair
2.8820^8m/s
Muhammed
which is correct answer
Vikash
he is correct but we can round up in simple terms
vijay
3×10^8m/s
vijay
is it correct
Fauzia
I mean 3*10^8 m/s ok
vijay
299792458 meter per second
babar
3*10^8m/s
Neyaz
how many Maxwell relations in thermodynamics
vijay
how we can do prove them?
vijay
What is second law of thermodynamics?
Neyaz
please who has a detailed solution to the first two professional application questions under conservation of momentum
Kwaku Reply
I want to know more about pressure
Osei
I can help
Emeh
okay go on
True
I mean on pressure
Emeh
definition of Pressure
John
it is the force per unit area of a substance.S.I unit is Pascal 1pascal is defined as 1N acting on 1m² area i.e 1pa=1N/m²
Emeh
pls explain Doppler effect
Emmex
solve this an inverted differential manometer containing oil specific gravity 0.9 and manometer reading is 400mm find the difference of pressure
Abayomi Reply
Einstine claim that nothing can go with the speed of light even its half (50%) but in to make antimatter they they hit the sub atomic particals 99.9%the speed of light how is it possible
Salima Reply
nothing with physical properties. this doesn't include things like particles and gravitational waves
Mustafa
that particles are of very small mass.... near equals to massless
Aritra
but they exist
vijay
yes they exist but mass is too less
Aritra
ok
vijay
greet all
Abayomi
the unit of radioactivity is .....?
Neyaz
Great Sharukh ! Do you have question in physics?
Bibekbir Reply
book says that when wave enter from one medium to another its wavelenght changes but frequency not how ? and f is inversely related to wavelenth
Sharukh
yes but how comes
Sani
how are you?
Sharukh Reply
please help me
World
what's the problem
Aritra
I really don't know physics.. I need help,in solving
Amara
me too
Ewulum
hii
Cheeru
I really don't know physics.. I need help,in solving
Cheeru
me too
True
I can teach u if u are ready
latunde
yes I am ready
True
hi
Emeh
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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