Demise of the Errant Balloon

An errant balloon, at an altitude of 1,000 feet, has eluded its captors for a considerable time. The balloon floats due east and over 2,000 yards away from a rifleman.  He realizes that his rifle had a range of only 1,600 yards; yet he took aim, fired and succeeded in shooting down the balloon.  Where and how could this occur?

a.     At high latitudes where the effect of Coriolis force is greater and extended his range.

b.     At ultra-high latitudes where the trajectory was along a great circle.

c.     At ultra-high latitudes where the trajectory was along a rhumb line.

d.     At the equator where the rhumb line and great circle are the same.

The answer is:

The first thought should be to recognize that a great circle is the shortest distance between two points on the surface of a sphere.  A rhumb line is the longer distance between two points.  This event can occur only at a location very close to the pole, where for short distances, there is a pronounced advantage in distance saved by a great circle over a rhumb line (in this case 23 per-cent) as seen in Figure 16 .  The trajectory of a bullet is along a great circle.  The due east direction of the balloon is measured along a small circle (a rhumb line), in this case the same latitude for both rifleman and the balloon, which intercepts all longitudes at a constant angle.  The great circle is consistently shorter than a rhumb line, but this advantage is not significant until longer distances are traversed.  The great circle between two points on the Earth is the arc intercepted at the surface of the Earth by a plane passing though the two points and the center of the Earth.  It is the smaller arc of the circle intercepted on the Earth.  The direction of a great circle measured in the Earth’s coordinate system is constantly changing, yet referenced in inertial space it is a straight line.

In the past, the great circle was difficult to employ in navigation despite its utility as its use required recalculating the great circle course repeatedly.  The rhumb line is easier to use on a Mercator chart as it appears as a straight line between two points and represents a constant course.  With the advent of computers and the inertial navigation system, the computation of the great circle course was easily accomplished and enabled a craft to steer or fly the shortest path between two points.

Great Circle and Rhumb Line Distance in the Polar Region

The conditions for this event are shown as follows:

Rifleman at 89.993°N latitude, 70.0°W longitude

Balloon at 89.993°N latitude, 70.25°E longitude

Great circle distance 1599.98 yards

Rhumb line distance 2082 yards

Rhumb line distance (on a small circle) is calculated by this computation:

cos latitude x longitude difference = 0.0001221 x 140.25 deg x 60 nmi/deg x 2,025.378 yds/nmi = 2,082 yards rhumb line distance

Since the location of this event is less than 0.5 nmi of the pole, the spherical triangle to be solved can be treated as a plane triangle shown in Figure 17 with little error incurred.  The law of cosines is typically used to calculate great circle distances and its formula is shown at the end.  The law of cosines is difficult to use when small angles are involved.  The value of cosine 1 arc-minute is 0.9999999 and the value of cosine 10 arcminutes is 0.9999957, which illustrates how the value of the cosine function for small angles changes very slowly.  The distance between the rifleman and the balloon, we will find, is 1,599.98 yards or 0.7899668 nmi or 0.7899668 arcminutes—an angle too small to obtain from the law of cosines formula (from a typical calculator).

To solve for the distance, we will use the plane triangle shown in
Figure 17
and employ the law of sines.

A  location of rifleman         AC   =   BC = 90°- 89.993° = 0.007°

B  location of balloon                  =   0.007° x 60 nmi = 0.42 nmi

C  pole                                                   

AB = 0.7899668 nmi x 2025.378 yds/nmi = 1,599.98 yds great
         circle distance

Figure 17.  Plane Triangle Solution to the Great Circle Calculation for the Errant Balloon

Distance:  Along the equator, the difference in longitude in arcminutes; along the meridian, the difference between latitudes in arcminutes.  Each arcminute is a nautical mile.  Formula:

cos D = sin(lat1)sin(lat2) + cos(lat1)cos(lat2)cos(Dlong)

Where        D  = Distance
               lat1  = Departure latitude
               lat2  = Destination latitude
            Dlong  = Difference in longitude between departure and
                        destination

The great circle is important for being the shortest distance between two points and because radio wave transmissions, natural phenomena such as light and many trajectories all travel along this path.

Afterword

A more precise solution can be obtained with the use of programs such as MATLABTM.