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110 2. Introductory applications

(a) Write your solution, and the differential equation, in terms of the non-
dimensional variables and introduce the
parameter Suppose that the limit of interest is (which
you may care to interpret); find the first two terms of an asymptotic ex-
pansion, valid for (in the form directly from the
governing equation. (You should compare this with the expansion of the
exact solution.) From your results, find approximations to the time to reach
the maximum height, the value of this height and the time to return to the
point of projection.
(b) A better model, for motion through an atmosphere, is represented by the
equation

where is a constant. (This is only a rather crude model for air
resistance, but it has the considerable advantage that it is valid for both
and Non-dimensionalise this equation as in (a), and then write
where Repeat
all the calculations in (a).
(c) Finally, in the special case (i.e. the escape speed), find
the first term in an asymptotic expansion valid as Now find the
equation for the second term and a particular integral of it. On the assump-
tion that the rest of the solution contributes only an exponentially decaying
solution, show that your expansion breaks down at large distances; rescale
and write down—do not solve—the equation valid in this new region.
Q2.28 Earth-moon-spaceship (1D). In this simple model for the passage of a spaceship
moving from the Earth to our moon, we assume that both these objects are fixed
in our chosen coordinate system, and that the trajectory is along the straight line
joining the two centres of mass. (More complete and accurate models will be
discussed in later exercises.) We take x(t) to be the distance measured along this
line from the Earth, and then Newton’s Law of Gravitation gives the equation
of motion as

where m is the mass of the spaceship, and the masses of the Earth
and Moon, respectively, G is the universal gravitational constant and d the
distance between the mass centres. Non-dimensionalise this equation, using d
as the distance scale and as the time scale, to give the non-
dimensional version of the equation (x and t now non-dimensional) as