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Vectors in Physics
Applications

We live and move about on the surface of a spinning sphere from which we can observe heavenly objects in the three-dimensional space that surrounds our planet. Most human movement is confined to the Earth's surface and its thin air space. Even airplanes follow curved paths, more or less parallel to the Earth's surface.

When we travel we must find our way from a departure point to a destination. To accomplish this goal, most of us either follow predetermined marked paths, roads and such, or we rely on professionals, bus drivers and airline pilots, to get us from here to there. Occasionally we get a taste of the navigation challenge, when we hike in unfamiliar surroundings or pilot a boat on featureless waterways. It is in such circumstances that we appreciate that to avoid "getting lost" we need to know where we are relative to our destination.

Mapping out the features of the Earth and giving travelers the tools to determine their position has been a long and tedious task that involves the talents of numerous people in diverse lines of work, from navigators and astronomers in ancient times to physicists, electronic engineers, and computer specialists in more recent times. Any time the airplane you travel in lands safely, even though the visibility is barely enough to allow the pilot to taxi to the gate, you reap the benefits of the hard and creative work of the navigation pioneers.

To see how far we have come in this endeavor consider the fact that a view of any spot on the Earth from a satellite is available for free on the Web. Click the image of the Earth on the left and see for yourself.

Specifying a Location on the Surface of the Earth

Latitude and Longitude

Twenty-two hundred years ago the Greek scientist Hipparchus proposed the clever coordinate system that we still use today. The two perpendicular directions needed to establish such a system on the two-dimensional surface of the Earth are the North-South direction and the East-West direction.
At any point on the Earth imagine a straight line drawn from the center of the Earth out into space. If the Earth were a perfect homogeneous sphere, that would be the vertical, the line defined by a plumb bob at the surface of the sphere. Hipparchus proposed that the angular separation between two such lines drawn at two locations along a circle passing through the poles (a great circle) be used as one of coordinates. Taking the equator as the origin, the angles so defined specify the LATITUDE of a location on the Earth. All the points sharing the same latitude fall on a circle called a PARALLEL. Parallel circles get smaller and smaller as the latitudes approach the poles.

To specify a displacement in the East-West direction, the angle between two great circles passing through the poles will do the job. The great circles passing through the poles are called MERIDIANS. The meridian passing through a place on the Earth determines its LONGITUDE. After much haggling it was decided in 1884 that the zero meridian, or the prime meridian, passes through the Greenwich Royal Observatory near London. Thus every point on the Earth is now uniquely defined by its latitude and its longitude. For example, the latitude of the San Francisco airport is 37° 37' N, and its longitude is 122° 23' W.
The meridians differ from the parallels in two important ways. First, there is no natural zero meridian, the role played by the equator for the set of parallels. Second, unlike the parallels, the meridians turn once a day with respect to the fixed stars. This last fact makes it much more difficult for a traveler to determine the longitude of a place than its latitude. More about this in the astronomical navigation section below.

Finding Our Way

Dead Reckoning

The first travelers facing serious navigation issues were the sailors who ventured out on open waters beyond the landmarks. Similar problems were encountered by travelers in the deserts where landmark features are also sparse.

These early navigators developed a procedure called dead reckoning.

Navigation by dead reckoning involves keeping track of the speed and direction and accurate knowledge of the displacement vector between the departure and destination points. Usually it is impossible, or at least inconvenient, to travel along that displacement vector. For various reasons, being subject to wind or currents in the sea for example, some segments of the trip depart from the direct displacement vector. Keeping track of the segments and simple vector addition allow the navigator to reach the destination successfully. In antiquity the North Star and later the magnetic compass were useful tools to determine the direction. Speed was measured in various ways, some more accurate than others. Dead reckoning was, and still is, a learned skill. Christopher Columbus sailed across the Atlantic mostly by dead reckoning. He was an acknowledged master of the method.

Astronomical Navigation

As the range of navigation expanded from coastal waters to the open seas and new uncharted territories, the importance grew of knowing the position accurately. In the seventeenth century, European rulers issued decrees distributing overseas territories according to their location (with complete disregard for the local inhabitants of course).

Suddenly it became very important that a captain of a ship determine the latitude and the longitude of a foreign destination he has just reached so he can claim it for his sovereign.

Latitude was easy. The position of the Sun at noon will easily yield the latitude. Consider the diagram on the left. At noon during the equinox the sun is exactly vertically overhead at the equator. At other latitudes the Sun is sighted at an angle equal to the latitude of the place away from the vertical. Thus a sighting of the Sun will yield the latitude. It is not too difficult to accomplish the same feat at other times of the year and using other celestial bodies. Obviously a lot of astronomical work goes into the effort, but sighting alone will do the job.

Determining the longitude is a much more difficult task. The meridians don't stay in the same place relative to the celestial sphere. Suppose a traveler were to compare some configuration in the heavens to the same configuration at the home location. As the Earth turns, the configurations look different at different times. To make a meaningful comparison, the traveler would have to know the exact time of day.
The determination of longitude was such a serious problem that it eventually involved virtually all the scientists of the day. In 1666 the Royal Academy of Sciences was established in France to tackle the problem. Eventually it was realized that the problem would not be solved until somebody invented a timepiece that would keep accurate time, even on a moving ship.
The search for longitude was also vigorously pursued in England. In 1714 the British parliament established a prize of 10,000 pounds for the longitude within one degree and 20,000 pounds for the longitude within 30 seconds. The watch to do the job was invented and built in 1728 by John Harrison. Once the longitude and the latitude could be accurately determined, commercial navigation progressed rapidly.

Navigation by Satellite

The most recent addition to the arsenal of navigation tools is the Navstar Global Positioning System known as GPS for short.

GPS was developed for military navigation in the '70s. It has since been made available to everybody. The GPS system consists of twenty-four satellites in various orbits, ground stations that control the satellites, and relatively inexpensive GPS receivers to help everyone from hikers and fishermen to truckers and pilots find their way. Each satellite continuously broadcasts its position and the current time. The signals from the satellites travel at the speed of light and reach a GPS receiver at slightly different times. Starting with the known position coordinates of the satellites and the differences between the times the signals were transmitted and received, the GPS receiver can calculate its own position.
First-generation GPS launches started in 1978. The new, second-generation GPS set became operational in 1995.

Inertial Navigation

Imagine traveling in a conveyance, say an airplane, equipped with a device capable of determining the acceleration (magnitude and direction) as a function of time. Also on board is a computing device which can calculate, from the acceleration data, the velocity as a function of time, and from the velocity, the position as a function of time. Such devices exist. They are called inertial navigation systems (INS). Travel with an inertial navigation system is completely self sufficient. It needs no connection to the outside world. It does not need to see the stars, nor does it need to send and receive radio signals. The heart of INS is the gyroscope, studied later in this course. A gyroscope is a spinning object, mounted in such a way that it has three degrees of freedom. The laws of physics dictate that left to itself, a gyroscope will maintain its direction in space. As the mounting mechanism of the gyroscope changes its orientation in space in response to the accelerations experienced by the vehicle, sensors on the gyroscope record the change of the mounting relative to the orientation in space maintained by the gyroscope. This information is fed into the onboard computer, which can thus keep track of the acceleration vector as a function of time. Integrating the acceleration data into velocity and position vectors gives all the information the moving vehicle needs to navigate.

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Further Study Questions:

1 . What is the longitude angle between two spots on the Earth whose local times are an hour apart?

To create paragraphs in your essay response, type at the beginning of the paragraph, and at the end.

2 . What is the distance at the equator between two spots on the Earth whose local times are an hour apart? What about at your latitude?

To create paragraphs in your essay response, type at the beginning of the paragraph, and at the end.

3 . If all airplanes carried GPS receivers, could we do away with air traffic controllers?

To create paragraphs in your essay response, type at the beginning of the paragraph, and at the end.

4 . What is the smallest number of satellites needed to make GPS work? Why do we have twenty-four of them?

To create paragraphs in your essay response, type at the beginning of the paragraph, and at the end.

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