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200 5. Some worked examples arising from physical problems

where each for
Now we suppose that h = O(1) as then the time at which this is attained
is also O(1), and hence our asymptotic expansions are valid for In
particular, for z = h and selecting the larger of the roots for i.e. further away, we
find (from (5.6)), that

where provided that (For the case of the shot-put
application of this model, the landing point is below the projection point, so h < 0
and this condition is certainly satisfied.) Finally, using this asymptotic expansion for
in (5.5), we find that the range is

from which, for example, we can estimate the angle which maximises the range; this
is left as an exercise.

This example has proved to be particularly straightforward; indeed, because of the
specific application that we had in mind—the shot-put—all the asymptotic expansions
are uniformly valid. On the other hand, if we had projected the object from the top
of a high cliff, then we would encounter the problem of i.e. and
then the validity of the original expansions would be in doubt. The expansions are
not valid when but we still have u = O(1) although this
investigation is also left as an exercise.

E5.2 Child’s swing
We are all familiar with the child’s swing, and the technique for increasing the arc
(i.e. the amplitude) of the swing. The process of swinging the legs (coupled with a