6. Two particles are moving with a constant speed v such that these are always at a constant distance d apart and their velocities are equal and opposite. After what time do these return to their original positions

Answer 1

Speed=v (in given units)
magnitude of velocity of each particle = v
direction: opposite

Velocity of particle 1: <v cosθ,v sinθ>
velocity of particle 2: <-v cosθ -v sinθ>

Distance between particles:
√(((vcosθ-(-vcosθ))²+((vsinθ-(-vsinθ))²)
= 2v = constant

Thus the particules are in circular motion and are diagonally opposite, with radius of circle = speed v.

The time to return to the original position
= circumference of circle / speed
= (2πv)/v
= 2π (unit depends on the time unit of v).