Question

A plane, moving with a speed of 500 knots, flies the great circle route from Prestwick (4◦ 36` W, 55◦ 30` N) to Gander (54◦ 24` W, 48◦ 34` N).What distance does it cover and how long does the flight take?

Answer 1

Answer on Question #54949, Physics / Astronomy | Astrophysics

Let A and B represent Prestwick and Gander respectively in figure so that the great circle arc AB is the flight route:

If the meridians PBD and PAC are drawn from the north pole P through B and A to the equator DC, triangle PBA is a spherical triangle. Applying the cosine formula, we may write:

$$\cos AB = \cos AP \cos BP + \sin AP \sin BP \cos APB$$ $$BP = 90{}^{\circ} - 48{}^{\circ}34' = 41{}^{\circ}26' = 41{}^{\circ}4333$$ $$AP = 90{}^{\circ} - 55{}^{\circ}30' = 34{}^{\circ}30' = 34{}^{\circ}5000$$ $$< APB = 54{}^{\circ}24' - 4{}^{\circ}36' = 49{}^{\circ}48' = 49{}^{\circ}8000.$$

Although the data are given to an accuracy of 1 arc minute, it is advisable to carry one or two extra figures to avoid rounding-off error vitiating the results.

The calculation then proceeds:

$$\cos AB = \cos 34{}^{\circ}5000 \cos 41{}^{\circ}4333 + \sin 34{}^{\circ}5000 \sin 41{}^{\circ}4333 \cos 49{}^{\circ}8000$$

giving $\mathrm{AB} = 30{}^{\circ}7061 = 30{}^{\circ}42'$ to the nearest minute.

Hence,

$$AB = 30{}^{\circ}42' = 1842'$$

The distance AB is, therefore, 1842 nautical miles.

$$\text{Time taken} = \frac{\text{distance}}{\text{speed}} = \frac{1842}{500} \text{hr} = 3^h68 = 3^h41^m0$$

Answer: $3^h41^m0$

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