Two bodies, c1 and c2, begin to move from the same point at constant angular velocity, but in opposite directions, along a 30 m radius circumference. If the first one takes 20 seconds to complete a rotation and the second takes 60 seconds, calculate:

a) The time that they take to meet.
b) The angle and distance traveled by each one.



R = 30 m
T1 = 20 s
T2 = 60 s
φ01=φ02= 0 rad

Since their trajectory is a circumference and their angular velocities are constant we face a problem of uniform circular motion or u.c.m.

Question a)

Since we know the period of each one, we can calculate the angular velocities by means of the following equation:




Both bodies will meet each other before completing a rotation. In particular, when they meet the sum of their angular positions will be exactly 2π radians.




If we use the equation of position of φ1 , and substitute, we get:

φ1=φ01+ω1·t 2π-φ2=0+ω1·t 

Knowing that similarly φ2=ω2·t, then:

2π-φ2=ω1·t  2π-ω2·t=ω1·t (ω1+ω2)·t=2π π10+π30·t=2π t=15 s

Question b)

If we replace the time at which they meet (t = 15 s) and the angular velocity in the equation of position of φ1, we get that the angle traveled by c1, which is:

φ1=π10·15 φ1=1.5π rad

And that the angle traveled by c2 is:

φ2=2π-φ1φ2=2π-1.5π φ2=0.5π rad

To calculate now distance traveled (s) for each of the two bodies, simply apply the equation s=φ·R:

s1=φ1·R=1.5π·30 = 45π ms2=φ2·R=0.5π·30 = 15π m

Formulas worksheet

These are the main formulas that you must know to solve this exercise. If you are not clear about their meaning, we recommend you to check the theory in the corresponding sections. Furthermore, you will find in them, under the Formulas tab, the codes that will allow you to integrate these equations in external programs like Word or Mathematica.