# Problem with relative motion in two dimensions

1. Sep 24, 2010

1. The problem statement, all variables and given/known data
Ship A is located 4.8 km north and 3.0 km east of ship B. Ship A has a velocity of 22 km/h toward the south and ship B has a velocity of 40 km/h in a direction 37° north of east.
(a) What is the velocity of A relative to B (Express your answer in terms of the unit vectors i and j where i is toward the east.)
(b) Write an expression (in terms of i and j) for the position of A relative to B as a function of t, where t = 0 when the ships are in the positions described as above.
(c) At what time is the separation between the ships least?
(d) What is that least separation?

2. Relevant equations

Don't really think there are any relevant ones here - it's just vector addition and subtraction.

3. The attempt at a solution

Well, I got the answers to the first two parts.
The answer to a) is v = -31.9454i -46.0726j.
The answer to b) is r = (3 - 31.9454t)i + (4.8 - 46.0726t)j.

I'm completely lost on how to do the 3rd and 4th parts though - I get that I have to solve the above equation for t, but I have no idea how to do it.

2. Sep 24, 2010

### aim1732

Shift your observer to one of the ships. The other ship should seem to be coming towards the observer with a velocity as calculated. The minimum separation is then the length of the perpendicular on the path of the other ship from the ship where the observer sits.Time can then be calculated.
A little geometry will be needed too regarding the perceived direction of the other ship and initial separation.