Question

Two sea ports are on the same parallel of latitude 42◦ 27 N. Their difference in longitude is 137◦ 35 . Ship A and ship B sail at 20 knots from one port to the other. Ship A sails along the parallel of latitude; ship B sails the great circle route connecting the two ports. Calculate the time difference in their arrival times if they leave port together.

Answer 1

Latitude = 42°27' N

AP = BP = 90° - 42°27' = 47°33' = 47.55°

Longitude difference = 135°35' = 135.583°

APB = 135°35' = 135.583'

the parallel route is denoted by the red arc ARB and the great circle route is denoted by the yellow arc AYB. If the meridians PAC and PBD are drawn from the north pole P through A and B to the equator CD, triangle PAYB is a spherical triangle. Applying the cosine formula, we may then write

cos AYB = cos AP cos BP + sin AP sin BP cos APB

Substituting the values of AP and APB

We get

cos AYB = 86.177066°

AYB= 86.177066°

= 5170.624'

=5170.624 nautical miles

Time taken for the Great Circle Route = 5170.624/20 = 258.5313 hours

For parallel route,

The circumference of the parallel at latitude 42°27´ N = ,
where r = R cos AOC, AOC = 42°27´ and R = radius of the Earth = 3443 nautical miles.

The red arc ARB in Figure covers only a fraction of this circumference, where the fraction is given by AQB/360° and AQB is given by the difference in longitude of A and B. So,

Distance covered by parallel route =

= 6011.703 nautical miles

=6011.703/20 = 300.5851 hours

Difference in arrival times = 42.054 hours (using scientific calculator values)

= 42°3'14.163"

= 42 hours 3 minutes 14.163 seconds

= 1 day 18 hours 3 minutes 14.163 seconds

Scientific values

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