Page:Optimum height for the bursting of a 105mm shell.pdf/8

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Using the foregoing formulae the following table was derived.

TABLE III

The Maximum Range for the Effectiveness of Fragments of Various Masses
m in lbs.	range in ft.	m in lbs.	range in ft.
0.0005	17	0.025	551
0.001	40	0.030	630
0.002	73	0.040	769
0.004	119	0.050	895
0.006	156	0.060	1006
0.008	209	0.080	1202
0.010	253	0.100	1376
0.015	368	0.200	2046
0.020	466	0.400	2801

In deriving Table III, we have, as we have already stated, used the British results on the retardation in air and the penetration into wood of fragments. However, it is now held that Welch's formula underestimates (possibly by a factor $2/3$ ) the retardation in air. On the other hand it appears that using Welch's results both for the penetration in air and in wood we are led to consistent results. Thus Inglis using a different criterion for effectiveness (namely that a fragment to be effective must have 150 ft. lbs. of energy) and using a retardation factor in air which is two thirds of Welch's finds ranges of the order of 70, 110 and 280 ft. for masses 0.0023, 0.0040 and 0.0095 lbs., respectively. These ranges of Inglis should be compared with the values we have derived namely 80, 120 and 250 ft., respectively. The agreement is satisfactory.

4. The estimate of the optimum height. The estimate of the optimum height was made on the basis of the following model.

Let the shell explode at a height $h$ from the ground. We shall suppose further the lateral spray is symmetrical with respect to the $yz$ plane. Then the surface density of the fragments which arrive at the point $(x,y)$ and which are effective is given by

(4) $\rho (x,y)={\frac {1}{4\theta }}\int _{m_{*}}^{\infty }N(m)\,dm\,\,\cdot {\frac {h}{(h^{2}+x^{2}+y^{2})^{3/2}}},$