Question

Two particles, 1 and 2, move with constant velocities `v_1` and `v_2` along two mutually perpendicular straight lines toward the intersection point O. At the moment `t=0` the particles were located at the distances `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?

Answer 1

At time `t`,

`S^(2)=(d_(1)-v_(1)t)^(2)+(d_(2)-v_(2)t)^(2)`
`(d(S^(2)))/(dt)=2(d_(1)-v_(1)t)(-v_(1)t)(-v_(1))+2(d_(2)-v_(2)t)(-v_(2))=0`
`t=(d_(1)v_(1)+d_(2)v_(2))/(v_(1)^(2)+v_(2)^(2))`
`S_(min)=(d_(1)v_(2)-d_(2)v_(1))/(sqrt(v_(1)^(2)+v_(2)^(2)))`