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.
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?
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
- What are the shapes of the solutions?
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
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.