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 to reach 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 `barv_(BA)`
`varv_(BA)=barv_(B)-barv(A)`

Here, `|varv_(BA)|=sqrt((20)^2+(20)^2)=20sqrt2km//h`
i.e, `barv_(BA)` is `20sqrt2` km/h at an angle `45^(@)` from east towards north. Thus, the given problem can be simplified as

A is at rest and B is moving with `bar_(BA)` in the direction shown in figure. Therefore, the minimum distance between the two is
`S_("max")=AC=ABsin45^(@)`

`=10(1/(sqrt2))km=5sqrt2km`
and the desired time is
`t=(BC)/(|barv_(BA)|)=(5sqrt2)/(20sqrt2)(BC=AC=5sqrt2km)`
`=1/4h=15min`