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Intercepting ICBMs in Three Dimensions

This Demonstration simulates an Intercontinental Ballistic Missile (ICBM) being chased by an Extended Range Interceptor (ERINT). Three sample paths are given for the ICBM: logarithmic (most typical), helical, and a combination of the two. The ERINT pursuit is governed by the iteration of differential equations in a manner similar to that of Euler's method. See Details for more complete discussion.

SNAPSHOTS

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DETAILS

The sliders on the left adjust parameters for the velocities of the ERINT and ICBM as well as the initial , , and coordinates of the ERINT. The initial position of the ICBM is fixed for each function set. The "predict" slider adjusts the ERINT's path so that it aims at where the ICBM is headed number of seconds into the future. At low values, this can often lead to a slightly more efficient interception path, but predicting too far ahead can cause the ERINT to overshoot the target. Checking the "maximum efficiency interception" checkbox causes the ERINT to determine the point on the ICBM's path that minimizes time to interception assuming the ERINT must travel at uniform trajectory. This turns out to be the fastest possible path of interception given any set of initial conditions. With the "performance" settings, you can adjust the number of points that are calculated and plotted by varying the iteration time interval, you can specify the time before the pursuer gives up pursuit, and you can turn the tail plots on or off.
The ICBM's path is determined by a set of three equations, one for each coordinate. The equations are mapped to curves in three dimensions by iterating the functions until an intercept occurs. Simply inputting a time variable as the parameter for equation creates the desired curve, but the ICBM will not travel at a constant speed, since the points will not necessarily be equidistant. To ensure that the points are equidistant, and thus that the ICBM will travel with constant speed, we first solve the equation
for , where is the value that will make the coordinate a constant distance of from . The equations are then plotted as functions of , and constant speed is guaranteed.
The ERINT's path is a repeated iteration performed over the interval with respect to the following differential equations:
where is the distance between the two objects at time . Note that this is very similar to Euler's method for systems, except that here we are finding a change in distance for each coordinate, rather than an instantaneous slope.
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