[From Interior Ballistics]
Let MN (see Figure) represent the burning surface of a powder grain (smokeless or otherwise). From this, as shown in the figure, issues a stream of powder-gas normally. Denote the pressure of the powder-gas at the instant of formation by F, and the pressure of the gaseous surrounding medium, outside the issuing streams of gas by P. Also place p equal to the varying pressure in the stream, and p equal to the variable mass of unit volume, in it. Suppose the cross-section of the stream to be unity, and assume anywhere in it a lamina of thickness ds and at right angles to it. The measure of dp, the accelerating force of the lamina, will be (mass X acceleration)
The initial velocity of the issuing gas is o, and we will denote the final velocity by Vl.
Integrating between proper limits, combining constants in one when possible, and remembering that the velocity increases as the pressure decreases.