Two cars `A` and `B` are running with same speed `v`. A along east and `B` towards north. Car `A` is at origin `O` and `B` is at distance `d` as shown. After how much time, the distance between cars is minimum and what is the minimum distance?

Explanation

First, find the velocity of `B` w.r.t. `A`


`v_(B//A)=sqrt(2) v`
`alpha=45^(@)`
Now, `A` is at rest and car `B` is moving on line `KP`.

Drop a perpendicular on line `KP` from `O`.
`OD`: minimum distance between cars
`OD=d sin 45^(@)=d/(sqrt(2))`
The time taken by car `B` to move from `K` to `D`
`t=(KD)/(sqrt(2)v)=(dcos45^(@))/(sqrt(2)v)=d/(2v)`
Alternatively
At time `t`, we have

Distance beween the cars
`s^(2)=(vt)^(2)+(d-vt)^(2)`
For `s` to be minimum
`(ds)/(dt)` or `(d(s^(2)))/(dt)=2vtxxv+2(d-vt)(-v)=0`
`vt=d-vt implies t=d/(2v)`
`s_(min)^(2)=(vxxd/(2v))^(2)+(d-vxxd/(2v))^(2)`
`=(d/2)^(2)+(d/2)^(2)=(d^(2))/2`
`s_(min)=d/(sqrt(2))`