Overhead Victoria West (S31°25' E023°06') at 09:15 UTC, en route to Cape Town International (S33°58' E018°36'), with W/V 180°/15 kt and TAS 120 kt. Calculate the ETA for Cape Town International.
Refer to figures.
To calculate the ETA, we first need to calculate the distance and groundspeed (using track, TAS, and W/V) between the two locations to determine the time. From the coordinates, the aircraft is traveling in a south-westerly direction. However, since both latitude and longitude are changing, we must account for both components.
STEP 1: CALCULATE THE DISTANCE OF CHANGE IN LATITUDE
1. Calculate the change in latitude (ch.lat):
- For different hemispheres, add the two latitudes.
- For same hemispheres, subtract the smaller latitude from the larger one.
Since both positions lie in the Southern Hemisphere, we will subtract the latitudes. Therefore,
- Change in latitude = S33°58' - S31°25'
- Change in latitude = 2°33'S*
* We are travelling towards ‘South’.
2. Calculate the distance:
A change of 1° of latitude corresponds to a distance of 60 NM. Therefore,
- Distance (NM) = ch.lat (°) x 60 NM
- Distance = 2°33' x 60 NM
- Distance = 153 NM
STEP 2: CALCULATE THE DISTANCE OF CHANGE IN LONGITUDE
1. Calculate the change in longitude (ch.long):
- For different hemispheres, add the two longitudes.
- For same hemispheres, subtract the smaller longitude from the larger one.
Since both positions lie in the Eastern Hemisphere, we will subtract the longitudes. Therefore,
- Change in longitude = E023°06' - E018°36'
- Change in longitude = 04°30'W*
* We are travelling towards ‘West’.
2. Calculate the distance:
The length of 1° of longitude varies with latitude. Therefore, the ‘departure’ formula is used to calculate the east–west distance.
Mean latitude: (Lat 1 + Lat 2) / 2 = (S33°58' + S31°25') / 2 = S32°41'30''
- Departure (NM) = ch.long (°) x cos (mean lat) x 60
- Departure = 04°30' x cos (32°41'30'') x 60
- Departure = 227 NM
STEP 3: CALCULATE THE ROUTE DISTANCE
Using Pythagoras' Theorem:
- Hyp2 = Base2 + Perpendicular2
- (Route distance)2 = (Distance of ch.lat)2 + (Distance of ch.long)2
- √(Route distance)2 = √[(153 NM)2 + (227 NM)2]
- Route distance = 274 NM
STEP 4: CALCULATE THE TRUE TRACK
From the figure, a right-angled triangle is formed. Therefore,
- tan (angle) = Perp / Base
- tan (angle) = Distance of ch.long / Distance of ch.lat
- tan (angle) = 227 NM / 153 NM
- tan (angle) = 1.48
- angle = arctan (1.48)
- angle = 56°
Since the track is measured clockwise from True North. Therefore,
- True Track: 180° + 56° = 236°
180° is the angle from True North to the base, and the angle between the base and the hypotenuse is 56°.
STEP 5: CALCULATE THE GROUNDSPEED
Given the following information, we can now calculate the groundspeed using a flight computer (CRP-5)
- W/V: 180°/15 kt
- TAS: 120 kt
- Track: 236°
The calculated groundspeed is 111 kt.
STEP 6: CALCULATE THE ETA FOR CAPE TOWN INTERNATIONAL
With the groundspeed and distance known, we can now apply the formula to calculate the flight time to Cape Town International. Therefore,
- Time (hr) = Distance (NM) / Groundspeed (kt)
- Time = 274 NM / 111 kt
- Time = 2.47 hr or 2 hr 28 min
The ETA is: Overhead Victoria West time + flight time to Cape Town International = 0915 UTC + 2:28 = 1143 UTC
The closest answer is: 1144 UTC
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