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
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).
To determine the time it takes for the particles to return to their original positions, we can use the concept of relative velocity.
Let's assume the two particles are moving along a straight line, and particle A is ahead of particle B initially.
Since the velocities of the particles are equal and opposite, the relative velocity between them is the sum of their individual velocities, which is 2v.
The relative distance between the particles is d, and they need to cover this distance to return to their original positions.
To calculate the time required, we can use the formula: time = distance / velocity.
Using this formula, we have:
time = d / (2v)
Therefore, the particles will return to their original positions after a time of d / (2v).