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Trajectories of small horizontal velocity in a resisting medium

Published online by Cambridge University Press:  24 October 2008

J. E. Littlewood
J. E. Littlewood
Affiliation:
Trinity College, Cayley Lecturer
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Extract

The results of the present paper were developed some years ago for quite trivial purposes. It seems possible, however, to judge by a recent publication, that they may sooner or later have scientific utility. They have also a certain slight intrinsic interest.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 22 , Issue 3 , 20 September 1924 , pp. 217 - 233
Copyright
Copyright © Cambridge Philosophical Society 1924

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References

* Richardson, L. F., “Theory of the measurement of wind by shooting spheres upwards,” Phil. Trans. Roy. Soc., A, 223 (1923), 345–382.CrossRefGoogle Scholar

† The corresponding problem for the ascending branch can be dealt with by a parallel argument involving only systematic changes throughout (and of course new tables).

‡ μy is a “pure number.”

§ Also the parallelism of argument in obtaining formulae in terms of a given t would have been less apparent.

* We have and

* It must be remembered that in these formulae ø is a function of z and a.

* The number z, from now on, only occurs as the apparent variable in an integration from 0 to p.

* For the logarithm in ø2 is greater than log ½ (algebraically), and less than

while

† The term is a maximum when when it is O (a ½).

* For if then and if then

† The approximation kx = Uψ is given by L. F. Richardson, loc. cit.

* The integral can be found in finite terms by known methods.

* That is, ignoring the term (which corresponds to about − 4 feet in x).

† The X 2 table was computed by a step-by-step process. It can be checked at ψ = ¼π, where it is correct, and at ψ = ½π, where The computation process gave 1·162, and the results of the table may be wrong by +·003 between 45° and 90°.

* Probably ⅔ ≤ β ≤ 1 in the general case.

* For example, to obtain E in powers of t we may work out, in powers of E and t′, the height y of the particle above the plane at any time t′. Expressing that this is zero we have an equation connecting E and t, soluble by successive approximation.

Doing this both for constants (μ, λ) and we obtain by comparison a formula for or β. The method is similar in the other problems considered. The more elaborate cases are exceedingly laborious, though the results are simple.

† To avoid possible misunderstanding it should perhaps be said that the value of β is not indifferent to a first approximation; any constant value other than 0 gives an error of lower order (larger) than that resulting from β = 0.

* In §§ 3.7, 3.8 we have only considered limiting values of the β's, where only the first power of λ is relevant, and the law of density might equally have been taken as 1 − λy.

As regards the law of resistance f (v) = μv n the only change in the general case is an additional term in the formula for N below, n being interpreted to mean Vf′ (V)/f (V).

† This is to suppose that λ V 2/g is comparable with unity. In the actual atmosphere λ V 2/g is about unity if V = 1000 f.s.

* To take an analogous example: certain formulae give the range of trajectories of ø = 15° with a proportionate error less than 10−4, although the velocity may fall to about ⅖ of its original value.