Page:Optimum height for the bursting of a 105mm shell.pdf/7

3. The effectiveness and the ranges of the fragments of different masses: 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 $m$ (in lbs.) will be effective if it has a velocity greater than $v_{*}(m)$ where

(1)$v_{*}(m)=210\,m^{-1/3}$

where $m$ is expressed in lbs. and $v$ in ft/sec.

Now the velocity with which the fragments are shot off from a 105mm shell is of the order of 3500 ft/sec. Consequently, a fragment of this shell which has a given mass $m$ will be effective only if it has traversed a range less than a certain value $r(m).$ To determine this maximum range as a function of the mass of the fragment the following formulae (again based on Welch) were used:

$r(m)=s_{1},\,\,v_{*}(m)=3500\,\,e^{-s_{1}/776\,m^{1/3}},$(2)

${\text{if}}\,\,v_{*}(m)\geq 1100\,\,$

and

$r(m)=s_{1}+s_{2},\,\,$(3)

$v_{*}(m)=3500\,\,e^{-(s_{1}+s_{2}/3)/776\,m^{1/3}},\,\,$

$1100=3500\,\,e^{-s_{1}/776\,m^{1/3}}\,\,{\text{if}}\,\,v_{*}(m)<1100.$