What Direction Should Ship A Steer to Minimize Distance to Ship B?

[thepopasmurf]

I'm studying for an exam and this is one of the sample questions which I cannot figure out.

Homework Statement

Two ships A and B move with constants speeds 48 km/h and 60 km/h respectively. At a certain instant ship B is 30 km west of A and is traveling due south. Find:

i) The direction ship A should steer in order to get a close as possible to ship B.

Homework Equations

V_ab = V_a - V_b

The Attempt at a Solution

Well, V_b = 0i - 60j
V_a = -48cosxi - 48sinxj
and therefore
V_ab = -48cosxi + (60 - 48sinx)j

I have drawn diagrams to help but I can't figure out the right angle. I tried differentiation to find the lowest possible value for j (they are already horizontally across from each other) but I got 1 for an answer which I know is not correct. Any advice.

 

[The Bob]

Have you tried relating each 'line' between objects as vector?

The Bob

 

[thepopasmurf]

I'm sorry, I'm not sure what you mean.

 

[The Bob]

When you drew your diagrams, you must have had a series of velocity vectors relating to the velocities of the ships. By using x = vt you could also obtain position vectors for each of the lines in your diagram. Doing this you could use normal addition of vectors to find your unknown, in this case the closest distance between A and B, and find an equation to be solved for it.

As this may still be confusing then let me give you my first line and see if that helps:
x_AB = -v_Bt + 30i + v_At

where x_AB is to be the distance between A and B, v_A is the velocity vector of A, v_B is the velocity vector of B, 30i is the distance between A and B initially, t is time and I have assumed that the closest distance between A and B must be on the same trajectory as A will initally head to get to this point.

That make any sense whatsoever?

The Bob

 

[thepopasmurf]

Bob, Your method I think will give the closest distance given a direction. (I think, haven't studied it yet). I don't know the direction. But I solved it. Using a relative velocity diagram you can derive a formula which relates the path (angle) of A to its angle. Then the cure all for all these max/min questions. Calculus. You can differentiate the values (sorry for being vague) and get the answer.

 

[The Bob]

Far enough. It would be hard, I admit, but possible to solve with the two unknown variables. However, if you are happy with your (far-easier-it-should-have-been-staring-us-in-the-face) method then cool. Glad you've got it and that I was no help whatsoever.

The Bob