Two particles, 1 and 2 move with constant velocities `v_1` and `v_2` along two mutually perpendicular straight lines towards the intersection point O. At the moment `t=0` the particles were located at the distance `l_1` and `l_2` from the point O. How soon will the distance between the particles become the smallest? What is it equal to?

Explanation

After time .t. ,the position of the point
A and B are `(l_(1)-v_(1)t)` and `(l_(2)-v_(2)t)`. Repectively.
The distance L.between the points A. and B. are
`L_(2)=(l_(1)-v_(1)t)^(2)+(l_(2)-v_(2)t)^(2)`
From minimum value of `L(dL)/(dt)=0`
`(v_(1)^(2)+v_(2)^(2))t=l_(1)v_(1)+l_(2)v_(2)` or `t=(l_(1)v_(1)+l_(2)v_(2))/(v_(1)^(2)+v_(2)^(2))`
Putting the value of t in equation (1)
`L_(min)=(|1_(1)v_(2)+l_(2)v_(1)|)/(sqrt(v_(1)^(2)+v_(2)^(2)))`