Two ships are sailing in the fog and are being monitored by tracking equipment.

Question

Two ships are sailing in the fog and are being monitored by tracking
equipment. As they come onto the observer's rectangular screen, one,
the Andy Daria (AD), is at a point 900mm from the bottom left corner
of the screen along the lower edge. The other one, the Helsinki (H),
is located at a point 100mm above the lower left corner along the left
edge. One minute later the positions have changed. The AD has moved to
a location on the screen that is 3mm west and 2mm north of its
previous location. The H has moved 4mm east and 1mm north. Assume that
they will continue to move at a constant speed on their respective
linear courses. Will the two ships collide if they maintain their
speeds and direction? If so, when? If not, how close do they actually
come to each other?

Explanation / Answer

a) No they would never collide because the AD crosses the H path at a different time interval.

b) Write the points AD and H as vectors that change over time. In this way we can use the distance formula, ((x2-x1)2+(y2-y1)2), using the AD as (x2,y2) and H as (x1,y1) and time as t:

     AD=(900-3t,0+2t)

     H=(0+4t,100+1t)

     Distance as a function of time=(((900-3t)-(0+4t))2+((0+2t)-(100+1t))2)

                                                 =((900-7t)2+(-100+t)2)

     Plugging the graph into the calculator shows the distance of each ship from the other at each moment of t. Since the graph never touches the x axis, the ships never crash. The closest the ships come to each other is 28.28427 mm at t=128.

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