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?

Answer

A, B

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)`
`=20sqrt2 km//h`

i.e. `v_(BA)` is `20sqrt2 km//h` at an angle of `45^@` from east towards north. Thus, the given problem can be
simplified as A is at rest and B is moving with `v_(BA)` in the direction
shown in fig
Therefore, the minimum distance between the two is
`s_(min)=AC=AB sin45^@`
`=10(1/sqrt2) km`
` = 5sqrt2 km`
and the desired time is
`t=(BC)/|v_(BA)|=(5sqrt2)/(20sqrt2) (BC=AC=5sqrt2 km)`
`=1/4h=15 min`