At the $2023$ EAA Air Show in Oshkosh,

Wisconsin, over $100,000$ aircraft fly in$.$

Pilot's need to keep a watch out for

other aircraft. As an example, three

Cessna $172$ aircraft are flying at the

same altitude following a straight and

level course. At time $(t-0)$ seconds, the

distance $AD,BD,$ and $CD$ are $20km,25,$

and $28km,$ respectively. Airplane $(A)$

maintains a constant speed of $300$

$k\frac{m}{h}r$ and airplanes $(B)$ and $(C)$

maintain a speed of $400k\frac{m}{h}r.$

Determine:

a$.)$ The position and velocity of aircraft

$(B)$ relative to aircraft $(A)$ at time

$=3$ minute. Repeat for aircraft $(C)$ relative to aircraft $(A)$ at time $(t=3)$ minute.

b$.)$ The distance $({}_{AB})$ between aircraft $(B)$ and aircraft $(A)$ at time $(t=3)$ minutes. Repeat

for the distance $({}_{AC})$ between aircraft $(C)$ and aircraft $(A)$ at time $(t=3)$ minute.

c$.)$ The distance and time when the separation between aircraft $(B)$ and $(A)$ is a minimum.

Repeat to find the minimum separation distance and corresponding time between

aircraft $(C)$ and aircraft $(A).$

d$.)$ If aircraft $($A$)$ collides with either aircraft $(B)$ or $(C),$ then calculate the speed for aircraft

$(A)$ so that aircraft $(A)$ flies midway between aircraft $(B)$ and $(C).$