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foimer curves, since these are plotted for unit air density. When the retardation for any velocity is found from the first curves, it is multiplied by the air density corresponding to the height at which the projectile is then located.
The air-density curve is plotted from the equation of densities found by Bessel to satisfy refraction data, as shown in Chauvenet's discussion of atmospheric refraction, and corresponds closely to similar data assumed by Alger and Ingalls.
Suppose that at a known point in the air, the direction and amount of motion, or the velocity V, of the projectile are known. The retardation caused by the air friction will be in the direction of V, at that instant, and its amount is given by the curves. The velocity V can be resolved into vertical and horizontal components, Vv and V\, and likewise the retardation r at this point can be similarly resolved into rv and rh, these latter being in the same ratio to r as are Vv and Vh to V. .
After an indefinitely small time, dt, the velocity V has been reduced by air friction by the amount rdt. This can be resolved
into rh = rdt ^*, and rv — rdt ^rr > these being the horizontal and
vertical components, or the reductions in the horizontal and vertical velocities.
In the same interval of time, gravity causes a further reduction in the vertical velocity, considering a velocity upwards as positive, amounting to gdt.
The velocity horizontally has then, in the interval of time dt,
been reduced to l\ — rdt ^, and the velocity vertically has been reduced to Vv- (rdt Ytf +gdt).
The combination of these new components of the velocity by means of a table of squares or similar means gives a new velocity V along the tangent, whence a new value of r can be deduced.
If we take the mean of the two vertical velocities and multiply by dt, we get the vertical travel during this interval, and, likewise, from the two horizontal velocities, the horizontal travel, whence
the coordinates Y and X at the new point can be determined, since the coordinates at the original point are known.
In this way, step by step, we can find V, 1, X, and Y, by intervals of dt. This is the method of analytical geometry.
MECHANICAL INTEGRATION The interval dt is indefinitely small, and unless some method can be found for determining a soluble equation of motion, the analytical method is impracticable.
If, however, we take n successive intervals, each equal to dt, and if the successive values of r are r, r1, and r2, etc., and of V, Vh, and Vv, similar quantities with the proper suffixes, then after an interval of time ndt the horizontal velocity would be V. V Vhn dt, or V-ndt ·
and the vertical velocity would be
V.-ndt The time interval ndt is now a measurable quantity, and can be called At. The quantity her " is the mean value during the interval At of the horizontal component of the retardation r, the quantity er is likewise the means of the vertical component
of r, and 29 is the mean value of g during the interval. . During a definite time interval At therefore, the horizontal
velocity Vn has been reduced to Vn-rn'At, and the vertical velocity from Vy to Vo-rot-g'At, where rh', rú', and g' are the mean values of the quantities rh, rv, and g during the interval At.
Now, if the interval At is not too large, the mean values ru', ro', and g' are for practical purposes the same as the means of the values of rn, Tv, and g at the beginning and end of the interval At. The first set of values we know, since we know V, Vn, and Vo
The second set is found as follows, by the well-known method of successive approximations : ,
Having tabulated V h, Vv, V, r, rh, rv, and g for the first known position of the projectile, we estimate new values of rh, Yo, and g
after the interval of time At. The value of g changes very slowly and no great error is involved in assuming the new value the same as the old.
Having assumed the next values, the means of the two values of r», rv, and g are taken, and new values are found for V\ and Vv by subtracting from the first values of V* the mean of the two values of fh, multiplied by At, and from the first value of V„ the mean of the two values of rv, plus the mean of the two values of g, each multiplied by A*.
Having thus found approximate values of V\ and Vv at the end of the interval At, they are combined by a table of squares or similar means to give the new value of V. A new value of r is now taken from the curves, for the new value of V, and it is corrected for height by finding the value of 8 corresponding to the new height, and multiplying r by this value of 8. The new height is easily estimated, differing from the old height approximately by VvAt.
With the new value of r, corrected for height, we find new values of r» and rv. If the originally assumed values of these quantities had been correct, they would be the same as those now found. If not, we repeat the operation, using the new values found for r» and rv.
After one or two of these successive approximations, we find that the values of the quantities for the end of the interval remain unchanged. They are then the correct quantities.
Having now found Vn and Vv for the beginning and end of the interval of time At, the horizontal travel in the interval, or AX, is A* times the mean of the two values Vh, and the vertical travel, or AY, is At times the mean of the two values of Vv.
Knowing the original coordinates X and Y, we now find the new ones by applying AX and AY.
Sheet B shows a form used for working out a trajectory. The first values, for t=o, are the muzzle velocity for V, coordinates X and Y each zero (in the usual case), and other values derived from V.
These values being tabulated, new values of rk and rv are assumed and tabulated for the next interval. Without experience
or other guide there is not much accuracy to be expected in this first approximation.
The two values of rn are now averaged, and the average, multiplied by At, subtracted from Vn. Likewise, the two values of ru are averaged, and, after the value of g has been added to this average, the sum is multiplied by At and the product subtracted from Vv. Thus new values of Vand Vo are tabulated.
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These are now combined into a new value of V, by using a table of squares if the angle of departure is large, or by the use of a slide rule if it is small. In the latter case, V is found by increasing Vn by the quantity 1
We now consult the curves, and obtain for the value of v the corresponding value of r. Estimating the altitude of the projectile at the end of the interval, we also take the proper value of
8 from the curves, and multiply r by 8. The product is the proper
new value of r to be tabulated. This is now multiplied, by slide
V V rule, by -A- to get n, and by -jf- to get rv.
We now usually find a small difference between the assumed and derived new values of n and rv. The derived values are substituted, new averages of r* and r„ found, and Vh and Vv corrected. In most cases it is unnecessary to correct further.
The two values of Vh and Vv are then averaged, multiplied by At, and tabulated as AX and AF. These are then added to the previous values (in this case zero) of X and Y to find the succeeding values.
The process is then repeated for the next interval of time, and so on, until the value of Y again becomes zero, or the value that is finally wanted.
The remaining steps are obvious.
The last remaining correction is found by interpolation, the final value of X divided by 3 to change it to yards, and the value of AR, above described (see page 74), is subtracted to get the corrected value of R.
It will be noted that Af, the time interval, is 1 second. This simplifies the calculations considerably, and for ordinary trajectories it is sufficiently small. It is possible, for the longer trajectories, to use a larger interval without great error, and the amount of this error can be estimated. For smaller trajectories, for instance with small-arms, it would be necessary to use a shorter interval, in some cases as small as one-tenth second. This is because the value of r changes rapidly.
The values of all quantities used are determined to the nearest tenth of a unit when Af=i, except for V, which is determined to the nearest unit. Closer approximation is not necessary, if the old mathematical rule is followed that a fractional result is given to the nearest last significant figure used, and that if the fraction is one-half, the nearest figure is used which is even, not odd.
The columns of differences are very important, tending to control the accuracy and indicate errors before the work has gone too far. The differences at rh and rv are also important to determine the values of the next succeeding quantities.
The 20" slide rule is all that is necessary for this work, except that at angles of the trajectory greater than about 200 it is more