However, for the purpose of- the construction of ballistic tables, as distinguished from range tables, atmospheric conditions are assumed normal and trajectories are constructed with known values of C. In the construction of range tables by the use of ballistic tables or by direct calculation, changes in air density at the gun are accounted for by a factor A representing the density placed in the denominator of the expression for C, equation (3). and changes in form of head, yaw, etc., by the factor i, in that expression. As used here the term " yaw " means the divergence of the axis of the pro- jectile from the tangent to the trajectory, both on account of initial instability and of curvature of the trajectory away from the direction of the axis at a later period.
Example of Numerical Integration. To illustrate the manner in which equations (9) and (io) may be integrated numerically, we shall assume an example as follows:
Example I. A 155 mm. gun fires a projectile having a ballistic coefficient of 3-6, with an initial velocity of 2,400 ft. per second, at an angle of elevation of 30 degrees. To determine the elements of the trajectory, assuming normal atmospheric conditions:
The values of G and H are given in metres-per-second velocity and metres height respectively, so that all velocities and distances must be reduced to metres.
Initial Conditions. At the gun we have
v =2,400 ft. per second = 731-5 metres per second 0^ = 0=30
?' = 73 i "5 sin 30" = 365-8 y =o.
Since is 5351, the value of log. G(v) from the G table is 9-4515.
Since y is o, H(y) = I.
Placing logarithms in brackets, we then have p (9-45 ' 5 I0 )
a*
and
i? ,_ (9-4515-
3-6
E/= (9-45i5-io)X36 5 -8 =28 , 74
and
Ey' +2 = 28-74+9-81 =38-55 = -y".
At the start, then, the horizontal velocity of the projectile is decreasing at the rate of 49-78 metres per second and the vertical component of the velocity is decreasing at the rate of 38-55 metres per second.
First Interval First Approximation. If we take a small interval of time, we do not make any great error in assuming that the retarda- tions during the interval can be based upon the velocity and altitude at the beginning of the interval. Taking a J-second interval', the change in components of velocity is 12-4 and 9-6 metres respectively, making the velocities at the end of the first interval,
y' = 365-8- 9-6 = 356-2.
These velocities are lower than those that actually exist at the end of the interval, since the retardations are based on the components of the velocity at the beginning of the interval, and are consequently higher than the true average values during the interval. Using the velocity figures just obtained, we find the following values corre- sponding to the end of the first interval,
)' + ( 35 6-2)'.
100 100
._365-8 +356-2
2X4
G() =(9-4464-10) " ) = (9-9959-J
5120
,._
- go-2 metres
Ex'
(9-4464-10) (9-9959-10) X62i-r 3-6
(from equation (2).)
= 47-78
F , _(9-4464-io) (9-9959 -io)X356-2 T^
27-39
Ey'-f 2 = 27-39 +9-81 =37-20.
_ Second Approximation. The values of the components of retarda- tion at the beginning of the interval are based on the velocity at the beginning of the interval and are, therefore, higher than the average values during the interval. The values just obtained for the com- ponents of the retardation at the end of the interval are based on a velocity lower than the true one at the end of the interval and are, therefore, lower than the average retardation during the interval. Means between these two sets of retardation components are nearer the average values during the interval than either set. The retarda- tions for the J-second interval based on the mean values are,
49-78+47-78
2X4
38-55+37-20
2X4
12-2 = X"
- 9-5 = -y"
making the velocities at the end of the first interval,
y' = 356-3, and the altitude,
y =i
2X4
If we now take these values and recompute E x', E y'+g we find the values 47-80 and 37-21 respectively. In the average components of retardation during the interval, no essential change will be found, showing that by a second approximation we have reached a result sufficiently accurate.
Second Interval. -Beginning with the components of the velocity and the altitude of the projectile at the end of the first interval we may now proceed in like manner to determine the components of the retardation during the second J-second interval. However, we may shorten the work as we now know not only the values of the retardation components at the beginning of the second interval but also the amounts by which they have changed in the preceding J-second. If the same rate of change continues during the second interval we will have for the end of that interval,
Ex' =47-78 -(49-78 -47-80) =45-82 = -*" Ey'-g=37-2i -(38-55 -37-21) =35-87 =-y". I he corresponding velocities obtained by using the average retarda- tions during the interval as before are:
x' = 609-6, and y' = 347-2.
The altitude at the end of the second interval is y = 90 . 3 + 356-3+347-2 =
2X4
Using the last values and again computing retardation components we have
Ex' =45-94 E y+g = 35-98-
Velocities and altitude computed from these do not differ from the values obtained in the first approximation, showing that a second approximation is unnecessary in this case.
Continuation of the Process. Using exactly the same methods, it is possible to determine numerically, step by step, the values of y, x', y', x", and y". We might also determine x at each step, but it is' not needed in making the step-by-step calculations and is usually more conveniently determined by a summation of x' after all the other values have been determined.
Length of Interval. In the above example it was assumed that the change in x, y, x' or y' could be found by using the mean of the values of x', y', x" or y" at the beginning and end of the interval. To do this without making too large an error we must use a small interval or take account of second differences. The choice of length of interval will depend upon the ballistic coefficient, muzzle velocity and curvature of the trajectory at the point considered. If these, in combination, or separately, are such as to cause rapid changes in the components of the velocity or acceleration, a relatively short interval should be taken, as J-second in the examples above. Otherwise, the interval may be increased to J-second or longer, and when second differences are used, as will be explained below, to two or more seconds.
High velocities or low ballistic coefficients usually require smaller intervals than low velocities or high ballistic coefficients. It will in general be desirable to take a shorter interval at the very beginning of the trajectory than at a later period. In changing to longer intervals it is most convenient, in the computations, to take twice the interval just used. As the velocities increase in the descending branch of the trajectory it may be desirable in some cases to use shorter intervals again. If so, half the length of interval just used should be assumed.
Second Differences. The length of interval may be increased and the amount of computation materially reduced if second differences are taken into account in computing all of the functions of A as y. x', y', x" and y".
The following table shows the results of further computations on the example discussed above and gives first and second differences of y' for intervals of one second.
t
y
y'
jst'Diff.
2nd Diff.
o
I
2
o
347-5 661-2
365-8 329-8 298-1
36-0 31-7
4-3 3-5
3
269-9
In determining the value of Ay from the average vertical velocity for the interval between i = 2 and 2 = 3, we obtain, if we neglect second differences,
= 284.
The following figure showing y' plotted on an exaggerated scale, as a function of /. illustrates the error obtained if only first differences are used. The area of the figure between t = 2 and t=3 is the value of Ay determined by using first differences only, these two as well as other consecutive points on the y' line being connected by a right line.
