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The shortest distance between two points 011 the Earth (consideredto be a perfect sphere of radius R) is the distance along a greatcircle - the arc of a circle formed where a plane passing throughthe two points and the center of the Earth intersects the Earth' ssurface,

The position of a point on the Earth is specified by the point'slongitude ¢ and latitude /L Longitude is the angle between themeridian (a line from pole to pole) passing through the point andthe "prime" meridian passing through Greenwich U.K. Longitudeis taken to be positive to the east and negative to the west. Latitude

is the angle from the Equator along the point's meridian. takenpositive to the north.

Let the vectors from the center of the Earth to the two pointsbe rl and r2.The cosine of the angle e between them can be foundfrom their dot product, so that the great circle distance between thepoints is RO.

Find an expression for e in terms of the coordinates of the twopoints. Use a coordinate system with the x axis in the equatorialplane and passing through the prime meridian; Jet the z axis beon the polar axis, positive toward the north pole, as shown in thesketch.

z

North

t r-

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Measuring gThe acceleration of gravity can be measured by projecting 11 bodyupward and measuring rhe time that it takes to pass two givenpoints in both directions.

Show that if the time the body takes to pass u horizontal line A inboth directions is TA, and the lime to go by a second line B in bothdirections is Til. then, assuming that the acceleration is constant,its magnitude is

Height

SIls= TA2 - 1'82'

where h is the height of line B above line A,

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