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Why do this problem?

This problem is a difficult exercise in algebra, differentiation and trigonometry which draws together strands from polar coordinates and mechanics, although these are not necessary for the solution of the problem. The rich ideas covered lead to a genuinely beautiful result which is well within the reach of the more technically skilled 6th former.

Possible approach

The technical aspect of the transformation is well suited to an individual activity. The ideas concerning the derivation of the equation would lend themselves to class discussion, although this requires the knowledge of the acceleration of a particle moving in polar coordinates. Is the class working together able to derive the equation used in the process?

Key questions

Students should be enouraged to understand the variables in the problem before attempting to make the transformation
  • How do we make a change of variables in an equation?
  • What is the meaning of the variables used when rewriting the solution?
  • What are the shapes of the solutions?

Possible extension

Those who are keen to make an extension should be encouraged to derive the equation. Alternatively, they can use physical data for the earth and the sun to investigate how closely the solution obtained here matches reality (it is actually exceedingly accurate).

Possible support

Those stuggling could be given the solution and asked to change variables. Alternatively, they could be given the second form of the solution and asked to sketch the curves in each case. This is still a rich and interesting task in itself.