Question

It is a calm summer day in southeast Iowa at the Ottumwa air traffic control radar installation - except there are some small, locally intense thunderstorms passing through the general area. Only two planes are in the vicinity of the station: American Flight 1003 is traveling from Minneapolis to New Orleans is approaching from the north-northwest, and United Flight 336 is traveling from Los Angeles to New York is approaching from west-southwest. Both are on the path that will take them directly over the radar tower. There is plenty of time for the controllers to adjust the flight paths to insure a safe separation of the aircraft.

Suddenly lightning strikes a power substation five miles away, knocking out the power to the ATC installation. There is, of course, a gasoline powered auxiliary generator, but it fails to start. In desperation, a mechanic rushes outside and kicks the generator; it sputters to life. As the radar screen flickers on, the controllers find that both flights are at 33,000 feet. The American flight is 32 nautical miles (horizontally) from the tower and is approaching it on a heading of 171 degrees at a rate of 405 knots. The United flight is 44 nautical miles from the tower and is approaching it on a heading of 81 degrees at a rate of 465 knots.

a. At the instant of this observation, how fast is the distance between the planes decreasing?
b. How close will the planes come to each other?
c. Will they violate the FAA's minimum separation requirement of 5 nautical miles?
d. How many minutes do the controllers have before the time of closest approach?
e. Should the controllers run away from the tower as fast as possible?

The specific questions asked above are a guide to your work and suggestions of the directions to pursue. Your report must contain not just answers to questions but explanations as well.

Answer 1

here the angles are 81 and 171 degrees are taking into the consideration and these are 90 degrees away,

so x and y axis are considered,

for American " A "

and for United " B "

so A ( t ) = 32 - 405 t

B ( t ) = 44 - 465 t

the distance is f ( t ) = ( A(t)² + B(t)² )

= ( ( 32 - 405 t )² + ( 44 - 465 t )² )

a )

differentiation is

df ( t ) / dt =   ( 2 ( - 405 ) ( 32 - 405 t ) + 2 ( - 465 ) ( 44 - 465 t ) ) / ( 2 ( ( 32 - 405 t )² + ( 44 - 465 t )²)² )

initial,

( 2 ( - 405 ) ( 32 ) - 2 ( 465 ) ( 44 ) ) / ( 2 ( ( 32 )² +( 44 )² )² ) = - 614.271816759

here negative symbol says this is decreasing.

now,

b )

2 x ( - 405 ) ( 32 - 405 t ) + 2 ( - 465 ) ( 44 - 465 t ) = 0

solving for "t"

we get

t = 0.08788954635108

so f ( t ) is now

f ( 0.08788954635108 ) = ( ( 32 - 405 t )² + ( 44 - 465 t )²)²

= ( ( 32 - 405 x 0.08788954635108 )² + ( 44 - 465 x 0.08788954635108 )² )

= 4.7677417030230

c )

Will they violate the FAA's minimum separation requirement of 5 nautical miles -YES

according to the above result,

they violate the 5 nautical mile rule

d )

0.08788954635108 hours = 0.08788954635108 x 60

= 5.273372781065 mins

e )

the controllers run away from the tower as fast as possible - NO

if they run with 5 nautical miles also the above result is 4.7677417030230 away.