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is the angle of projection The small parameter is (>0), and for
motion such as that for a shot-put (see Mestre, 1991) its value is typically about 0.01. In
these projectile problems, the main interest is in estimating the range (and maximum
range), x, for a given vertical displacement (z), which may be zero; here, we find x
where z = h. We seek a solution, following the straightforward procedure (which may
produce a uniformly valid solution; cf. §2.3):

and then from (5.1a,b) we obtain

At the next order, we have the equations

which give, after an integration and on using the initial conditions on
t = 0,

Thus we have, for example,

which remains valid only if t is smaller than and at this stage we do not know
the domain for which z = h at
Finally, we integrate once again to find how the position of the object, (x, z), de-
pends on time (t), on and on