Question

Two particles, 1 and 2, move with constant velocities v₁ and v₂ along two mutually perpendicular straight lines toward the intersection point O. At the moment t = 0 the particles were located at the distances l₁ and l₂ from the point O. How soon will the distance between the particles become the smallest? What is it equal to?

Answer 1

Let the particle 1 and 2 be at points B and  A  at t=0 at the distance l1 and l2 from intersection point O.

Let us fix the inertial frame with the particle 2. Now the particle 1 moves in relative to this reference frame with a relative velocity vector (v₁₂=v₁-v₂) and its trajectory is the straight line BP. Obviously, the minimum distance between the particle is equal to the length of the perpendicular AP dropped from point A on to the straight line BP(Fig.).

The sought time can be obtained directly from the condition  that