Two ships A and B are 10 km apart on a line running south to north. Ship A farther north is streaming west at 20 km/h and ship B is streaming north at 20 km/h. What is their distance of closest approach and how long do they take tor each it?

Explanation

Ships A and B are moving with same speed 20 km/h in the directions shown in figure. It is a two dimensional, two body problem with zero acceleration.let us find `v_(BA)`

`v_(BA)=v_(B)-v_(A)`
Here `|v_(BA)|=sqrt((20)^(2)+(20)^(2))=20sqrt(2)` km/h
i.e. `v_(BA)` is `20sqrt(2)` km/h at an angle of `45^(@)` from east towards north. Thus, the given problem can be simplified as

A is at rest A and B is moving with `v_(BA)` in the direction shown in figure.
Therefore, the minimum distance between the two is
`s_("min")=AC=ABsin 45^(@)`
`=10(1/(sqrt(2)))km=5sqrt(2)km`

and the desired time is `t=(BC)/(|V_(BA)|)=(5sqrt(2))/(20sqrt(2))(BC=AC=5sqrt(2)km)`
`=1/4h=15` min