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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.

  • A

    1157 UTC

  • B

    1144 UTC

  • C

    1140 UTC

  • D

    1152 UTC

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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