Ã¢â‚¬Â¢ reference text:
Calter and Calter, Technical Mathematics With Calculus, 5th Edition
Answer all ten questions. Show your working at all stages, as marks will be deducted if your reasoning is not clear, and you will not be able to get more than half-marks if working is not shown.
Each question is worth 1 mark, for a total of up to 10 marks for the assignment.
1. Answer the following questions about calculating altitudes
a) A plane takes off, and needs to be at an altitude of 2500m when it reaches 30 km from the airport (ground distance). What (vertical) angle do it need to ascend at? How far has it actually flown to get to 2500m, given that it is heading upwards as well?
b) Assume the plane is now at an altitude of 2500m. If it stays at the current rate of ascent, how many km (by air and over ground) will it be from the airport when it reaches 4000m?
2. Answer the following questions about calculating groundspeed. For this question, ignore the altitude and calculate only the ground speed.
a) The plane has travelled 54 km North, then another 13 km West. How far over ground is the plane from the airport?
b) The plane has now travelled another 16 km North. Now what distance d it is from the airport?
c) What angle and approximate compass direction has the plane travelled from the airport? Express this in 'true' reckoning as well as a compass bearing.
3. Answer the following questions about emergency landings.
a) The plane needs to make an emergency landing. The pilots have a choice of two landing points: one due North 31 km away, the other is 45 degrees to the East from the plane's current position and it is 25 km away. The plane has a tail wind of 12kmh as it heads north but this will act as a cross wind if the plane heads East. The plane's range is only 32km (in air, not groundspeed), and its airspeed it 150kmh. Can it reach both landing points?
b) The pilots choose the landing point that is due East. If they head there immediately, what will their angle of descent be? Recall from question 1 that they are travelling at 5000m.
4. In each of the following graphs, convert the specified angles to bearings. Write them as both compass point bearings (e.g. N45ºW) and from true (e.g. 45ºT).
5. Using the law of sines, find the missing angles and sides in the following triangles
a) A = 5562, A = 25º 14', b = 38º 27.
b) a = 5, c = 6, A = 30º
6. Using the law of cosines, find the missing angles and sides in the following triangles.
a) a = 20, b = 8, C = 134º
b) a = 12, b = 19, c = 28
7. Calculate the effect of wind in the following scenarios.
a) A pilot is flying at a bearing of 250°T but there is a wind of 22kmh in the direction 160°T. If the airspeed is 172kmh, find the track and groundspeed.
b) A pilot is flying in the direction 45ºT. The wind is from due west with a speed of 36kmh, and the aircraft's airspeed is 300kmh. Find the track and groundspeed.
c) A pilot flies on a heading of 125ºT. His aircraft's airspeed is 220kmh and the speed of the wind is 30kmh. If the wind is blowing from 210ºT, find the track and groundspeed.
8. Answer the following navigation questions.
i) A flying doctor plane based in Birdsville has two visits to make to remote homesteads. On the first visit, the doctor flies 72km at an angle of ?ºT, then changes course by fº and flies another 45km to the second visit. The second homestead is 60km east and 35km north of Birdsville. The doctor then returns to Birdsville, flying b km.
What is the doctor's first bearing (?), how much did she change course after the first visit (f) and how far was the final leg back to Birdsville (b)?
ii) A small plane needs to make an emergency landing. There are two possible landing points, one being at N42ºW and the other at N65ºE. The pilot has information that states the two landing points are 7.25 km apart, and that the bearing of the second landing point is S88ºE of the first. How close is the plane from the two landing strips?
iii) A Dash-8 aircraft is flying in a straight line in Washington state at 13,000 feet. Mt Rainier is seen ahead, on an angle of 25º from the track of the plane. After travelling another 12km, the angle to Mt Rainier is now 65º from the aircraft's track. What is the closest the aircraft will come to Mt Rainier?
9. Perform the following navigational calculations using radians.
a) A plane is flying due South along the great circle along the 0º longitude from London to Valencia (ignore for now the very slight difference in their longitudes). London's latitude is 51º 30' N while Valencia's is 39º 28' N. Assume the radius of the Earth is 6336km and that the plane is flying at 10,000m altitude. Calculate the distance the plane will fly between the two cities.
b) A plane is flying along the great circle from Sydney to Los Angeles. The approximate distance along this route is 12,065km. What angle does the plane cover, calculated from the centre of the Earth?
10. Perform the following calculations using radians.
i) A satellite is visible directly overhead in Singapore, and is 120km above sea level. It orbits the Earth over the Equator. The angle between the satellite's first position and its position 6 hours later is . How far in km has the satellite moved in that six hours? Assume that Singapore is on the Equator and that the radius of the Earth is 6336km. You may use 3.14 as an approximate value of .
ii) How far in km has Singapore moved during the same six-hour period, due to the Earth's rotation?
iii) Assuming the satellite was directly overhead as viewed in Singapore at the start of the six hours, where is it positioned after the six hours has passed?