Minimum distance between two particles in 2D motion

Hello students. Today we will solve a classic kinematics problem from the IIT 1965 paper. We have two particles, P and Q, moving along two mutually perpendicular straight lines towards their intersection point O.

Let's visualize the scenario. Particle P is moving towards O with a constant speed v1, starting at distance d1. Particle Q is moving towards O with speed v2, starting at distance d2.

At any time t, the distance of P from O will be d1 minus v1 times t. Similarly, the distance of Q from O will be d2 minus v2 times t.

Since the paths are perpendicular, the distance D between them at time t can be found using the Pythagorean theorem.

To find the time when the distance is minimum, we can differentiate the square of the distance with respect to time and set it to zero.

Let's expand and differentiate the expression.

Dividing by minus 2 and rearranging the terms, we get v1 times d1 minus v1 squared t plus v2 times d2 minus v2 squared t equals zero.

By pulling out time t as a common factor, we find that t equals d1 v1 plus d2 v2 divided by v1 squared plus v2 squared. This answers part one of our question.