Page:Optimum height for the bursting of a 105mm shell.pdf/11

 5. : We have seen that the height at which it is most advantageous to have a 105mm shell burst is about 75 ft. for wide limits in the effective area of the target involved. The reason for this rough independence can be understood in terms of the following considerations:

Essentially the existence of an optimum height for burst depends on the extent to which the area on the ground where we are super-efficient (region I of §4) can be reduced. It cannot however be concluded that the optimum height corresponds to reducing this area to zero. For the fragments which are most numerous are those with small masses and these have ranges of the order of $$h$$ itself. Thus, when $$h=75$$, the fragments which are effective (i.e., with velocities greater than $$v_*$$ [Cf. eq. 1]) and which arrive on the ground directly below the point of burst must have masses greater than 0.0021 lb. (cf. Table III). Fragments with $$m$$ in this neighborhood are the most numerous, but their effect is confined to a very small area. Accordingly we must go to somewhat larger masses (with correspondingly longer ranges) to be really efficient. In other words, at the optimum height we must necessarily be super efficient in the regions directly below the point of burst. In other words the area over which we are super efficient must be reduced as much as is compatible with the circumstances of the problem.

The considerations of the preceding paragraph account also for relative insensitiveness of the optimum height with the target area within limits. For while a reduction in the effective target area requires a greater surface density of fragments this can be achieved by even a slight reduction in $$h$$ since by so doing we bring into range the smaller fragments which increase in number very rapidly with decreasing mass.

Again, it it clear that since a change in the half angle $$\theta$$ of the spray can be formally incorporated into the calculations as a change in target area, the optimum height is relatively insensitive also to changes in $$\theta$$. For a given total number of fragments, the surface density varies inversely as $$\theta$$. Hence, other things being equal a change in $$\theta$$ alters the surface densities by a constant factor and the final efficiency will be clearly the same if $$\theta$$ were kept unaltered but the target area increased by the same factor.

In our calculations we have assumed that the spray is symmetrical with respect to the $$yz$$ plane. A tilt of this spray with the vertical by a small angle $$\phi$$ can be taken into account by using for the optimum height the value

Considering all the uncertainties of the problem we may suggest as the optimum height for burst a value of 75 ft. In doubtful cases it might be safer to go down to as much as 70 ft. and in any event greater heights should be avoided as far as possible.