Relative Velocity

Question

Two aircraft A and B are at the same height and are travelling horizontally at 500km/h. A is flying due north and B is flying due west. The bearing of B from A is 15 and the distance AB is 10km.

(i) Find the least distance between the aircraft in their subsequent motion.
(ii)the time, in seconds , for them to reach the position where they are the least distance apart.

 

You can draw a vecotr diagram to help you out!

Answer 1

Here is how I would set this problem up. Choose B's initial position at the origin of a coordinate system (0,0).

Express B's position at time t (hrs) as (-500t,0) because its movement is due West.

 

Express A's initial position in the third quadrant such that you have right triangle ANB with hypotenuse 10 km, and its coordinates based on bearing angle NAB = 15^o and initial A(-10cos15^o,-10sin15^o).  N represents intersection of the northbound vector beginning at A with the westbound vector coming from B.

We have A's position at time t as A(-10cos15^o,500t-10sin15^o).

 

Obviously this is worthy of drawing & studying to make sure that it is sensible. If you decide that it is, then you simply should use the distance formula to express AB as a function of t. Then minimize that function by Calculus methods.