Motion of a Balloon
Let M = the mass of the balloons, car, netting, gas, passengers, &c., on starting.
vo = the capacity of the envelope of the balloon when full.
vo= the volume of gas at the pressure of the air introduced into the balloon before starting.
v= he volume (supposed less than Vo ) occupied by the gas at the height x.
po = density of the gas in the balloon on the earth.
p= density of the gas in the balloon at the height x.
xo = density of the air on the earth.
x = density of the air at the height x.
u= the initial upward velocity of the balloon (which is introduced for the sake of complete generality, but is always zero).
uo = the velocity (vertically upwards as all horizontal motion is ignored) at height x.
Then the equation of motion at any time previous to the balloon becoming completely filled is
The last term being due to the resistance of the air, which is assumed to vary directly as the square of he velocity and as the density of the air. In very slow motions these resistance appears from experiments to vary pretty nearly as the velocity; and when the motion is very swift, as in the case of a rifle-bullet, as the cube of the velocity; but when the motion is neither very rapid nor very slow, the law of the square of the velocity probably represents the truth very fairly. By g is denoted the value of gravity at the height x, so that
a being, as above, the radius of the earth. In the exponential term, we shall replace g by g, as no sensible error can result thereform. The value of xu is constant, as by Boyle's and Marriotte's law it always = xo vo. Writing, therefore, for brevity --
The equation of motion rakes the form
Whence, following the usual rule for the integration of linear differential equations of the first other, and writing X for e-nz, for convenience of printing,
Herein put x = 0, so that u = uo and we have
Whence, by subtraction,
In which Ei x I used to denote the exponential integral of z, viz.: FORMULA according to a recognized notation. The values of the integral Ei x, which may be regarded as a known function, have been tabulated (see Philosophical Transactions for 1870, pp. 367-388).
We thus have, except for temperature, the complete solution of the problem of the motion of the balloon so far as velocity and height are concerned; it would not be possible to connect the time and the height except by the performance of another integration, for the practicability of which it would be necessary to submit to some loss of generality, viz., we should have to regard x as small as compared to a, and take x as small, and so on. The equation last written gives the motion until the height (say h) is attained at which the balloon, becomes quite full, after which the gas begins to escape, and have the second case of the problem.
Before proceeding, however, to the discussion of this second case, it is worth while to examine the solution more carefully, leaving out of consideration quantities that make no very great difference in the practical result, for the sake of simplicity. Supposing, then, gravity to be constant at all heights, and x to be zero, the equation of motion takes the simple form.
And we see, what is pretty evident from general reasoning, that if a balloon, partially filled, rises at all, it will at least rise to such a height that it will become completely full.
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