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ContentsInitial Problem Statement 2 Narrative 3-14 Notes 15 Appendices 16-23
Wind VectorsHow does the crew of a moving ship determine the wind speed and direction and how is this information used by a helicopter pilot who needs to land on the ship?
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How does the crew of the moving ship determine the wind speed and direction and how is this information used by the pilot?
A ship is on a mercy mission to supply aid to
victims of an island volcanic eruption when
there’s an unexpected emergency on board. The
medical specialist has been taken ill and needs
to be replaced. The ship can’t stop or turn round
as time is critical so the replacement has to be
brought on board by helicopter.
The ship needs to report the wind direction so
the pilot can adjust his course to intercept the
ship. This information is also vital to help the
pilot control the helicopter as he performs a
landing on a moving helipad.
Wind VectorsInitial Problem Statement
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Narrative Introduction
Multimedia DiscussionThe animation Wind Vectors Animation is available to introduce this example.
If you were the pilot of the helicopter trying to land on the ship what factors do you need to take into account in order to achieve your goal? How do the factors affect your direction of flight?
MultimediaThe interactive resource Wind Vectors Interactive 1 is available to help demonstrate some of the factors and how they affect the direction of flight. The resource is in the form of a game where you try to land your helicopter on a ship.
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2. Calculating your heading vectorIn order to travel in a straight line from a starting point to a destination point you need to know in which direction you must head. This is called the heading angle! By navigation convention the heading direction is given in degrees clockwise from the direction of North.
θ
Figure 1.In the introduction you discussed the importance of knowing the speed as well as the direction of travel of the helicopter, the ship and the wind. This is important in determining how long the time of flight will be and therefore how far the ship can move in that time. The speed and direction of motion are both scalars, i.e. they only have a magnitude. They can be combined to give the heading vector, h, which has both magnitude and direction.
θ
h
hy
hx
Figure 2.
N
S
EW
0°
90°
180°
270°
heading angle
direction of travel
N
S
EW
180°
270° 90°
0°
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Discussion DiscussionThe heading angle is shown in the diagram. How does the heading vector in the diagram show a representation of the speed?
The heading vector shows the speed and direction of travel. Can you think of another, more common name for the heading vector?
DiscussionWhat are the units of the heading vector?
Activity 1The heading vector can be written as a column vector of the form
h =
hhx
y
where hx and hy are shown in Figure 4. Write down expressions for hx and hy in terms of the heading speed, h, and heading direction, θ. Given a speed and heading direction what are the components of the column vector h?
Activity 2Verify that the magnitude of the vector h gives the correct speed, h.
MultimediaThe animation Wind Vectors Heading Animation is available to support this example.
MultimediaThe interactive resource Wind Vectors Interactive 2 is available to test your understanding of heading vectors and their components. The application provides a random heading vector which you must first plot, then use the components to calculate values for the speed and direction.
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S
EW
heading angle
direction of travel
90°
180°
270°
0°
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3. Using the heading vector to calculate positionThe heading vector, h, gives the velocity of an aeroplane, helicopter, ship, etc. The vector can be written as a two component column vector,
h =
=
hh
hh
x
y
sincos
θθ
Activity 3A helicopter has a cruising speed of 140 mph and travels with a heading angle of 60°. Write down the heading vector giving values to 2 d.p.
Activity 4If the helicopter flies on a fixed heading for 30 minutes how far will it travel from the point of take off? How far to the north/south and east/west as measured from the point of take off will it be? Give values to 2 d.p.
θ
Figure 3.
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Activity 5 (challenging!)You want to fly your helicopter to meet up with a moving ship. At the point of take off the ship is 100 miles due north of your location and travelling due east at a speed of 28 mph. Your helicopter flies at 140 mph. Calculate the following (giving answers to 2 d.p.)• How long will it take to fly to the ship?• What is the distance vector you need to travel?• What heading vector should you make to meet the ship? Check the magnitude of
the answer matches the speed of the helicopter!• What is your heading angle?
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4. Accounting for the effect of the windYou have seen how the use of vectors can help to plan a heading to meet a moving ship. This works when the helicopter flies in the direction in which it is pointing. However, you found out in the introduction that wind can change this and blow you off course.
MultimediaThe interactive resource Wind Vectors Interactive 1 is available to help demonstrate how wind can affect the direction of flight. The resource is in the form of a game where you try to land your helicopter on a ship.
To look at the direction of travel the helicopter must take to travel to a known point you must take account of the direction vector of the helicopter and the direction vector or the wind. The vector sum or resultant will give the actual direction of travel:
Figure 4.
Activity 6The heading of the helicopter is the vector h and the wind direction is given by the vector w. Write an expression for the resultant vector, r, as a vector sum and as a sum of column vectors.
Discussion MultimediaQualitatively, how would you take the wind into account when solving the problem of meeting with a moving ship as described in the previous activity?
The interactive resource Wind vectors Interactive 3 is available to test understanding of the calculation of resultant vectors.
direction of wind, w
heading of aircraft, h resultant heading, r
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5. Finding the apparent wind speed and directionIn cases where the helicopter does not take off from an airfield, the wind direction and speed can be recorded by the ship and passed on to the helicopter to help navigation to the ship and to provide details of the conditions when it arrives. This is useful for landing or in providing forward information if a helicopter has to hover over a moving ship, for example.
The speed and direction of the wind can be found using an instrument called an anemometer.
MultimediaThe animation Anemometer Animation is available to introduce how an anemometer works.
If the ship is stationary the wind speed and direction is the same as that observed by a person standing on the land. This is called the true wind and is the same vector, w, that will push the helicopter off course. However, in general the ship will be moving.
Discussion DiscussionWhat effect do you think the movement of the ship will have on how a person standing on the deck perceives the wind?
You are standing on a stationary ship and there is no wind blowing. What is the true wind speed and direction?
DiscussionThe ship now moves at 1 ms-1 due north. What would you feel?
Activity 7The wind due to motion is denoted by the vector m. Write down an expression for m in terms of the heading of the ship h.
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Activity 8You are on a stationary ship. You notice that there is a wind blowing from west to east at 3 ms-1. The ship now starts to move at a constant speed of 4 ms-1 in the direction of north. What wind do you feel now? This wind is called the apparent wind and is denoted by the vector a.
Use the following diagram to show your working.
Figure 5.
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DiscussionIs the apparent wind of use to the helicopter approaching the ship?
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6. Reporting the true wind speed and directionThe apparent wind is the wind that is measured by the ship using, for example, an anemometer. You have concluded that the apparent wind does not help the helicopter navigate to the ship as in general this is not the same as the true wind which causes the helicopter’s flight to deviate.
Multimedia DiscussionThe Anemometer Animation is available to refresh your memory of how an anemometer works and the relationship between the true wind and apparent wind.
How could the ship determine and report the true wind direction?
Activity 9Write a vector expression for the true wind, w, in terms of the apparent wind, a, and the heading of the ship, h.
The reporting of vector components may not be the most convenient form for the ship to relay the information to the helicopter. An alternative is to give the direction angle and speed.
Activity 10Write down expressions for the true wind speed and direction angle in terms of the components of the true wind vector, w. Note, the angle should be reported as a number between 0° and 360°.
Activity 11Check your expression for the angle gives sensible answers for all possible wind vector directions. If your expression does not work explain why this is so and devise a set of rules that you can apply to obtain the angle for a wind that is blowing in any direction.
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7. A f inal challengeThis section is a challenge that builds on all the activities covered so far.
Activity 12You want to fly your helicopter to meet up with a moving ship. You have no weather information at the point of take off and are reliant on the ship providing wind information. At the point of take off the ship is 100 miles due north of your location and travelling due east at a speed of 28 mph. The ship measures an apparent wind vector (in mph) of
a =−−
165
Your helicopter flies at 140 mph. Calculate the following (giving answers to 2 d.p.)
• Use the apparent wind vector and ship’s heading to calculate the true wind vector.• What is the speed and direction of the true wind?• What is your heading angle you must fly in order to meet the ship?• What is the time of flight?
DiscussionCompare the results with the previous case where there was no wind blowing; heading angle = 11.54 °, time of flight = 0.73 hours.
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NotesWind directionThe convention used in this example is that the wind is shown as a vector indicating the direction in which it is blowing. A wind blowing to the east is therefore shown as a vector from left to right and will have a direction relative to north of 90°.
The actual meteorological convention for wind direction is in terms of where the wind has come from, so that the above would be called a westerly wind as it has blown out of the west.
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Appendix 1using the interactives
Wind Vectors Interactive 1This resource is available to help demonstrate some of the factors that affect the direction of flight. The resource is in the form of a game where you try to land your helicopter on a ship.
Figure 17.
The selection box at the top left allows you to choose a scenario. The default scenario is a stationary ship with no wind.
The green buttons at the bottom right of the screen control your helicopter’s direction of flight. The rotate buttons change the direction of flight. Note, they are set up so that a single click will change the angle by a small amount. If you hold the button down the helicopter will not rotate.
The “Go!” button starts the flight. The goal is to land on the landing deck at the front of the ship.
Figure 18.
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The other options for the game are
• Allowtheshiptomove.• Allowthewindtoaffectthedirectionofflight.• Allowtheshiptomoveandthewindtoaffectthedirectionofflight.
These are selected by clicking the appropriate selection at the top left of the screen.
Figure 19.
Moving ship, no wind
Figure 20.
Stationary ship, with wind
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Figure 21.
Moving ship, with wind.
Wind Vectors Interactive 2This resource is available to test the understanding of heading vectors and their components.
Figure 22.
You will be given a vector which you must plot on the heading chart by moving the line to the correct point and clicking the mouse. When you click on screen you will be told if you are right or wrong.
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Figure 23.
If you give a wrong answer you can try again by clicking on the chart. If you get the answer right you are invited to work out the speed and heading. Answers should be given to 2 d.p. Remember that the heading is the angle in degrees clockwise from north.
Figure 24.
The button at the bottom allows you to check your answer. If you get the answer wrong you can try again.
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Figure 25.
Remember the angle is in degrees clockwise from north
Figure 26.
After giving the right answer the reset button starts the activity again with a different heading.
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Wind Vectors Interactive 3This resource is available to test understanding of the calculation of resultant vectors.
Figure 27.
This resource has narration which can be switched off or on by selecting “Start activity without voice over” or “Start activity with voice over” at the bottom of the screen respectively.
The next page introduces the problem. Click “next” to continue.
Figure 28.
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The next page gives the heading and wind vector. You must calculate the resultant vector.
Figure 29.
The check button allows you to check your answer. If you get the answer wrong you will be told so and you can revise your answer. If you get the answer right the resultant is plotted on the screen.
Figure 30.
The reset button can be used to reset the activity with different heading and wind vectors.
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Appendix 2mathematical coverageUse trigonometry and coordinate geometry to solve engineering problems• Know and be able to use Pythagoras' theorem• Know the meanings of sine, cosine and tangent• Be able to find sine, cosine and tangent for any angle• Be able to use sine, cosine and tangent to find unknown sides and angles in right angled triangles• Solve simple equations involving trigonometric functions• Know and be able to use vocabulary and notation appropriate to vectors at this level• Be able to work with vectors• Be able to solve simple problems involving vector quantities by scale drawing and calculation
Use algebra to solve engineering problems• Know and be able to use vocabulary and notation appropriate to arithmetic at this level• Be able to evaluate expressions• Know how to write any number to a specified number of decimal places or significant figures, or
to some other level of accuracy• Know and be able to use vocabulary and notation appropriate to algebra at this level• Change the subject of a formula• Be able to solve linear equations• Be able to solve simultaneous linear equations• Know how to check an answer by substitution• Work with Cartesian (x, y) and polar (r, θ ) coordinates and graphs and convert between these
forms
