Uniform Circular Motion 441

and

Vy == i) sin (^ + c). (lib)

Integrating (11 a) and (lib) for the positions (x, y) of the particle attime / we find

x � ki = + (v/w) sin (^ + ^) (12a)

y - k2 = - (y/u) cos (^t + c), (12b)

where k\ and k^ are constants of integration.Note that (12a) and (12b) imply

{x - ki)2 + (y - k2)2 == O/^)2 = constant. (13)

Equation (13) says the orbit of the particle is a circle with radius Rgiven by

R = v/co. (14)

Since ^»==^ equations (14) and (8) imply that

a = vw == z;V2?, (15)

which is the usual equation for calculating the centripetal accelera-tion of a particle undergoing uniform circular motion.Note that equations (2) and (5) imply equation (13). In words this

result says that circles in acceleration and velocity space imply thatthe orbit in real space must also be circular. This appears to be asomewhat new way of viewing uniform circular motion. Incidentallythis viewpoint can be generalized. For example ellipses in accelera-tion and velocity space imply the orbit must be an ellipse. The sameis true for parabolas, hyperbolas, spirals, and certain other simplefunctions. No applications of these generalizations are known atpresent.

ARCHAEOLOGISTS BUILD THEIR OWN SITEA new kind of archaeological site�a mound of soil, bones and pottery�will be

constructed this spring by Cornell University students to help scientists 100years from now.Under the direction of Dr. Robert Ascher, students will build a mound 80 feet

long, 15 feet wide and five feet high. As they build, they will bury articles suchas bones, pottery, coins and textiles. Then each year, for the next 100 years, one-hundredth of the loaf-shaped mound will be sliced off and inspected for changes.Deformation of the mound, effects of plant growth and of rain and snow, as wellas of burrowing animals and soil changes will be noted. It is hoped the experimentwill benefit archaeologists studying the final portion of the mound 100 years fromnow.