Time, Speed, Distance: Circular Motion

In this article, we will discuss the concept of circular motion. Objects (cars etc.) or people generally move around a circular track at different speeds, and they begin from a particular point either in the same direction or in the opposite direction.
The basic objective of this concept is generally to calculate:
  • The time of meeting of people (running around the track) at the starting point again after they started.
  • Meeting for the first time anywhere on the track.
  • At how many different points do people meet while running on the circular track?
Type 1: Time to meet at the Starting point
Step 1: To calculate the time to meet at the starting point we have to first calculate the time taken by each of the persons to run one complete round at their respective speeds.
Step 2: Then take the L.C.M of these times. This LCM gives the time of their meeting again at 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. When will A and B meet at the starting point for the first time?
Solution:
The time of their meeting again at the starting point will be the LCM of 120/12 & 120/20 i.e. 10 & 6 LCM of 10 and 6= 30
So, after 30 mins these people will be together at the starting point.
To check whether your answer is right or not
Calculate the position of A and B after 30 mins.
After 30 minutes, A would have taken 3 rounds and B would have taken 30/6 = 5 rounds. So after completing 3 & 5 rounds they will be at the starting point.
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Type 2: Time to meet for the 1st time anywhere on the track
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 1st 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.
Type 3: To calculate the number of points at which the objects 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.
In case of 3 objects / people A, B, C
To calculate the time at which they meet for the 1st time
First calculate the following:
Tab= Time in which B overlaps A or A overlaps B
Tac= 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 1st meeting.
To calculate the time at which they meet at the starting point for the 1st time
First calculate the following:
Ta= Time taken by A to complete the entire circular path.
Tb= Time taken by B to complete the entire circular path.
Tc= 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 1st time.
Similarly, we can apply the same concept for more than 3 objects as well.
Let us solve some questions based on the above discussed concepts.
Solved Questions
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 1st 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 2nd 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 3rd 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 7kmph, 11kmph, and 15 kmph respectively in the same direction. How many points are there where they will meet on the track, including the starting point?
Solution:
Let A be moving at a speed of 7kmph, B at 11kmph and C at 15kmph.
A and B meet at (11-7) points = 4 points.
B and C meet at (15-7) points=8 points.
HCF of 4, 8 will be 4.
Hence, the three meet at 4 points.
For doubts, post your comments below and our experts will provide you with the solutions.
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