numerical analysis Word Problem: In the 1991 Gulf War, the Patriot missile defen

Question

numerical analysis

Word Problem: In the 1991 Gulf War, the Patriot missile defense system failed due to roundoff error. The troubles stemmed from a computer that performed the tracking calculations with an internal clock whose integer values in tenths of a second were converted to seconds by multiplying by a 24-bit binary approximation to one tenth:

0.110 0.000110011001100110011002

(a) Convert the binary number to a decimal. Call it x.

(You may use Maple convert command: > x:=convert(0.00011001100110011001100,decimal,binary)

(b) What is the absolute error in this number; i.e., what is the absolute value of the difference between x and 0.1 ?

(c) What is the time error in seconds after 100 hours of operation (i.e., |3,600,000(0.1-x)|)?

(d) During the 1991 war, a Scud missile traveled at approximately MACH 5 (3750 miles per hour). Find the distance that a Scud missile would travel during the time error computed in (c).

(On February 25, 1991, a Patriot battery system, which was to protect the Dhahran Air Base, had been operating for over 100 consecutive hours. The round-off error caused the system not to track an incoming Scud missile, which slipped through the defense system and detonated on Army barracks, killing 28 American soldiers.)

Explanation / Answer

Part a

Converting binary value to decimal

x = convert(0.00011001100110011001100,decimal,binary) = 0.099999904632568359375

Part b

absolute error = |(0.1 - x)| = 0.000000095367431640625

Part c

time error after 100 hours = 100 hrs * absolute error

100hrs = 3600000 tenths of a sec; absolute error = 0.000000095367431640625

time error = 3600000 * 0.000000095367431640625 = 0.34332275390625 secs

Part d

speed of missile = 3750 miles per hour = 3750 / 3600 miles per second = 1.041667 miles per second

distance travelled in time error = speed * time error

distance = 1.041667 miles/sec * 0.34332275390625 secs = 0.35763 miles = 0.575 kms

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