Subrahmanyan Chandrasekhar

“ Another factor which it is necessary to decide on before we can estimate the height at which it it most advantageous to have the shell burst is concerned with the range and the effectiveness of a fragment of a given mass.
We shall suppose that a fragment is effective if it has a sufficient velocity to penetrate an inch of wood. Using a formula due to Welch (who has experimented on the penetration of wood by fragments of various masses) we find that a fragment of mass (in lbs.) will be effective if it has a velocity greater than where
(1)
where is expressed in lbs. and in ft/sec. ”

“ Fragments with 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. ”

“ 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 itself. Thus, when , the fragments which are effective (i.e., with velocities greater than [Cf. eq. 1] ) and which arrive on the ground directly below the point of burst must have masses greater than 0.0021 lb. ”