Circular Motion

Given that two particles are moving in uniform circular motion with the same angular velocity in opposite directions, we can use the following reasoning to find the minimum time after which their velocities become orthogonal:
- Let the two particles start at from the same point. One particle moves anti-clockwise and the other moves clockwise along the same circular path of radius .
- Initially, both particles have the same angular velocity , but they are moving in opposite directions.
- The velocity of each particle at any given time is tangential to the circular path. Since they are moving in opposite directions, at some point their velocities will become orthogonal to each other.
To find the time at which the velocities become orthogonal: - The angle between their velocities will become (or radians) when they are orthogonal.
- The angular displacement of the particles will be for the first particle and for the second particle.
- The relative angular displacement between the two particles is .
At the point when their velocities are orthogonal: Solving for : Thus, the minimum time after which the velocities of the two particles become orthogonal is .

When a particle moves in a circular path, in half the period of revolution it covers a semi-circular path. Let’s consider the motion of the particle:
- The total distance covered by the particle in half the period is half the circumference of the circle.
Thus, the distance covered is: - The displacement, which is the straight-line distance between the initial and final position of the particle, is the diameter of the circle. Therefore, the displacement is: Hence, the displacement is and the distance covered is .