1. Suppose two planes start descending at the same time, one is directly above t

Question

1. Suppose two planes start descending at the same time, one is directly above the other and both dh dt follow the same differential equation, =-hi/3 Is there any possibility of a midair collision? Will the initially higher one ever get below the initially lower one? Develop two different arguments to support your conclusion, one based on the uniqueness theorem and one based on the fact this differential equation is autonomous and hence graphs of solutions are related to each in a particular way

Explanation / Answer

dh/dt = -h1/3

=> h1/3dh = -dt

integrating gives

=> (3/4)h4/3 = -t + c

now at t = 0, h = ho => c = (3/4)ho4/3

=> t = (3/4)ho4/3 - (3/4)h4/3

=> 4t/3 = (ho4/3 - h4/3)

=> 4t/3 = (ho14/3 - h14/3) ....... (1)

and 4t/3 = (ho24/3 - h24/3) ........(2)

for collision => h1 = h2 => now as ho1 and ho2 are different, planes can't collide.

also initially higher plane will always be higher.

=> though difference in distance will going to decrease with time

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