Two bodies simultaneously begin to move in a circle from one point in the same direction. The orbital period

Two bodies simultaneously begin to move in a circle from one point in the same direction. The orbital period of the first body is 2 s, the second body is 6 s. Determine the time after which the first body will catch up with the second.

Given:

T1 = 2 seconds – the period of revolution of the first body moving in a circle;

T2 = 6 seconds – the period of revolution of the second body moving along the same circle;

It is required to determine t (seconds) – the time interval after which the first body will catch up with the second.

According to the problem statement, both bodies move along the same circle. Let the length of this circle be C. Then the linear speed of the first body will be equal to:

v1 = C / T1, and the linear velocity of the second body will be: v2 = C / T2.

Let’s find the speed of the first body relative to the second body:

v = v1 – v2 = C / T1 – C / T2 = (C * T2 – C * T1) / (T1 * T2) = C * (T2 – T1) / (T1 * T2).

Then, both bodies will again be at the same point after a time equal to:

t = C / v = C / (C * (T2 – T1) / (T1 * T2)) = T1 * T2 / (T2 – T1) = 2 * 6 / (6 – 2) = 12/4 = 3 seconds …

Answer: the first body will catch up with the second body 3 seconds after the start of the movement.