Question

The rays OM , ON , and OP in the figure, coincide at time t=0. At the same instant of time OM and OP rotate in the plane about point O in opposite directions while ON remains fixed. With respect to ON , segment OM rotates at constant rate of 3° per second and OP , at 6° per second. What is the smallest positive value of t, in seconds, for which segments OM and OP will coincide? When t=30 min, how many more revolutions does segment OP complete than segment OM ?

Answer 1

OM and OP will coincide a second time after 120 seconds. You could visualize it as ray OP moving at 3 degrees/second relative to OM.
Using this visualization, we find that OM makes 30*60*3/360 rotations (minutes * 60 * rotational speed/360 degrees in a rotation), which is 15, and that OP makes 30*60*6/360 or 30 rotations, which is 15 more than OM.