Time, Speed, Distance: Circular Motion

If 2 people have to meet for the first time, the faster person has to complete one full round extra over the slower person. The faster person is ahead of the slower one right from the first minute only due to his speed being higher than the speed of the other and they both are moving in the same direction.
It can be said that when the faster is ahead of the slower by one full track length, he will be overtaking the slower person from behind. Now, at this very moment these people meet.
In order to calculate the time we can say that time of meeting
= (track length/relative speed)

Example: There is a track with a length of 120 meters and 2 people, A & B, are running around it at 12 m/min and 20 m/min respectively in the same direction. When will B overtake A for the 1^st time?

Solution:
Time of meeting = 120/(20 - 12) = 120/8 = 15min.
In order to visualize we can say that B covers 8 mtr/min extra over A. So when B covers 120 mtrs. extra he will overtake A from the behind and hence they both will meet.

The logic that operates behind calculating this is that if we divide the time of their first meeting at the starting point with the time of their first meeting anywhere on the track, we get the number of points at which these people would meet including the starting point.

Example: There is a track with a length of 120 meters and 2 people, A & B, are running around it at 12 m/min and 20 m/min respectively in the same direction. At how many points will A and B meet?

Solution:
Number of points = 30/15 = 2 points
This is inclusive of the starting point.

So far, we have solved the 3 types of problems taking 2 people A and B.
The immediate question that comes to our mind is what if more than 2 people are given?
How will we then solve the question?
The above concepts are applicable for more than 2 people/objects as well.

First calculate the following:
T_ab= Time in which B overlaps A or A overlaps B
T_ac= Time in which C overlaps A or A overlaps C
Then take the LCM of these two and you'll get the time of their 1^st meeting.

First calculate the following:
T_a= Time taken by A to complete the entire circular path.
T_b= Time taken by B to complete the entire circular path.
T_c= Time taken by C to complete the entire circular path.

L.C.M of these three times will give you the time at which A, B and C meet at the starting point for the 1^st time.

Similarly, we can apply the same concept for more than 3 objects as well.

Question 1: Two people P and Q start running towards a circular track of length 400 m in opposite directions with initial speeds of 10 m/s and 40 m/s respectively. Whenever they meet, P's speed doubles and Q's speed halves. After what time from the start will they meet for the third time?

Solution:
Time taken to meet for the 1^st time= 400/(40+10)=8 sec.

Now P's speed = 20m/s and Q's speed=20 m/s.
Time taken to meet for the 2^nd time= 400/(20+20) = 10 sec.

Now P's speed =40 m/sec and Q's speed = 10 m/sec.
Time taken to meet for the 3^rd time= 400/(10+40)=8 sec.

Therefore, Total time= (8+10+8) = 26 seconds.

Question 2: Three persons start running towards a circular track of 36 km length at the speeds of 7kmph, 11kmph, and 15kmph respectively in the same direction. How many points are there where they will meet on the track, including the starting point?

Solution:
Time taken for the three persons for their 1st meeting at starting point = LCM of 36/7, 36/11 and 36/15 which is equal to 36 hours.
Time taken for the three persons for their 1st meeting = LCM of 36/(11-7), 36/(15 - 7) and 36/(15 - 11) which is 36/4, 36/8, 36/4 which is equal to 36/4 = 9 hours
So there will be 36/9 = 4 points where they will meet.