When it comes to time, speed and distance questions circular motion is almost always a painful topic to deal with. While linear motion is easy to visualize and comprehend, circular is motion is confusing on various levels. In this article, I will try to make it simpler to understand circular motion along with a few good questions.

**Question types**

The basic premise involves two or more individuals traveling along a circular path at varying speeds. You will be given the speeds or the ratios of their speeds and the length of the circular track or the radius in most of the cases. You have to find either:

(a) The number of distinct points of meeting

(b) The time when the individuals meet again at the starting point

(c) In rare cases, you might have the meeting points handy but would have to find the possible speeds of one of the individuals (it becomes a pseudo-numbers problem then with factors involved)

**How do the objects move?**

Now, if you look at the figure, you will understand that both A and B are moving in the same direction at speeds of 3 m/s and 5 m/s respectively. Now, it is obvious that for every second that A and B are running, B gains 2 m on A. Once B races ahead of A, B needs to cover 60m more than A to meet her again. So, it can be understood that B will cover this 60m at the rate of (5-3)=2 m/s and then meet up with A after 30 seconds. Effectively, the distance covered by B will be 60 m more than that covered by A. This observation is very important in questions involving laps.

Similarly, if they are moving in opposite directions, when they meet for the first time, both of them would have covered a distance of 60 m together. The next meeting point will come when they have covered 60 m between them again and this process will repeat.

**Number of distinct points of meeting**

I won’t really expect a direct question from this concept but this is useful in certain questions which ask how distance has been covered in total at a certain meeting point. The concept is simple and there are two cases:

(a) If the two individuals are moving in the same direction and their speeds are in the ratio a:b such that a and b are co-prime, the number of distinct meeting points is given by |a – b|

(b) If the two individuals are moving the opposite direction and their speeds are in the ratio a:b such that a and b are co-prime, the number of distinct meeting points is given by (a+b)

The problem arises when there are more than two entities moving along the track. The solution? Simple. Take two at a time and then see the overlap. Generally, if there are more than two individuals traveling along the path, the number of meeting points will not be greater than 4. So, if you find yourself getting into a lot of numerical data, rest assured that either the question is not worth solving or you have missed an important bit in the question.

So, if two people are running in the same direction starting from the same point with speeds of 2 m/s and 6 m/s the number of distinct points of meeting will be |1-3| = 2. Understand that here we take only the basic ratio and not the original speeds.

If the above mentioned people are running in opposite directions starting from the same point, we get a total of (1+3) = 4 distinct points.

If 3 people A, B and C with speeds 2 m/s, 5 m/s and 8 m/s respectively start running along a circular track from the same point such that A and C are traveling in the clockwise direction and B is traveling in the anticlockwise direction then:

(a) At how many distinct points do A and B meet?

(b) At how many distinct points do A and C meet?

(c) At how many distinct points do B and C meet?

(d) At how many distinct points do all three of them meet?

The answers to the questions are: 7, 3, 13 and 1. Can you figure out how?

**Time after which they meet again at the starting point**

There is a simple formula you can apply to solve questions involving multiple entities. It does not depend on the direction in which the entities are moving. It simply requires you to know either the individual times to complete a full circle or the ratios of their distances and speeds. The formula is:

Time after which, two or more entities meet at the starting point is given by:

LCM (individual times taken to complete a lap along the circular track)

**Points to remember**

(a) Use LCM and HCF of fractions in case you are not able to get integral values for individual times

(b) All the distinct points of meeting are equally spaced along the circular track in terms of both distance and time and are covered in a definite manner

(c) To understand the order in which these points are covered, you would need to plot the points on a circle and study the movement. As soon as they meet for the first time, that point would become the starting position and you would have to move ahead the same distance that you had done in the first case

(d) The order in which the points are covered would repeat as soon as all the entities meet at the starting point. So, if there are 7 distinct meeting points, the starting, the seventh, the fourteenth meeting points and so on would be the same. Similarly the first, the eighth, the fifteenth meeting points would be the same

(e) A common error made during attempting questions in a hurry is that students tend to calculate the LCM of times taken when you are in fact required to find the first meeting point. Understand that there is a difference between the two

(f) If you know the number of meeting points and the time after which they will meet at the starting point again, you can get the time required to meet for the first time by simply using: (Time after which they will meet at the starting point)/(Number of meeting points)

(g) If the entities do not start from the same point, you have to find the first meeting point. As soon as you get the first meeting point, you can approach it in the same manner as you solve circular motion problems.

A few problems for practice. Let us know how you solved these:

Three students Arun, Barun and Kiranmala start moving around a circular track of length 60m from the same point simultaneously in the same direction at speeds of 3 m/s, 5 m/s and 9 m/s respectively. When will they meet for the first time after they started moving?

(a) 30 seconds (b) 60 seconds (c) 15 seconds (d) 10 seconds

After what time will the two persons Tez and Gati meet while moving around the circular track? Both of them start at the same point and at the same time. (CAT 1997)

- Tez moves at a constant speed of 5 m/s, while Gati starts at a speed of 2 m/s and increases his speed by 0.5 m/s at the end of every second thereafter.
- Gati can complete one entire lap in exactly 10 s.

In a 4000 meter race around a circular stadium having a circumference of 1000 meters, the fastest runner and the slowest runner reach the same point at the end of the 5th minute, for the first time after the start of the race. All the runners have the same staring point and each runner maintains a uniform speed throughout the race. If the fastest runner runs at twice the speed of the slowest runner, what is the time taken by the fastest runner to finish the race? (CAT 2003 leaked)

(a) 20 min (b) 15 min (c) 10 min (d) 5 min

A sprinter starts running on a circular path of radius r metres. Her average speed (in metres/minute) is πr during the first 30 seconds, πr/2 during next one minute, πr/4 during next 2 minutes, πr/8 during next 4 minutes, and so on. What is the ratio of the time taken for the nth round to that for the previous round? (CAT 2004)

(a) 4 (b) 8 (c) 16 (d) 32

A jogging park has two identical circular tracks touching each other, and a rectangular track enclosing the two circles. The edges of the rectangles are tangential to the circles. Two friends, A and B, start jogging simultaneously from the point where one of the circular tracks touches the smaller side of the rectangular track. A jogs along the rectangular track, while B jogs along the two circular tracks in a figure of eight. Approximately, how much faster than A does B have to run, so that they take the same time to return to their starting point? (CAT 2005)

(a) 3.88% (b) 4.22% (c) 4.44% (d) 4.72%

Sriram is running along a circular track such that after every lap the distance gets halved. Also the time taken gets reduced to 1/3^{rd} of the present value. What is the average speed of Sriram after five laps, if the time taken for the initial lap is 1 hour and the distance of the initial lap is 1 km?

(a)1.29 kmph (b) 1.38 kmph (c) 1.49 kmph (d) 2.36 kmph

Meghana and Deepti run in opposite directions from a point P on a circle with different but constant speeds. Meghana runs in the clockwise direction. They meet for the first time at a distance of 900 m in the clockwise direction from P and for the second time at a distance of 800 m in the anticlockwise direction from P. If Deepti is yet to complete 1 round, then the circumference of the circle is

(a) 1200m (b) 1250 m (c) 1300m (d) 1700m

We hope you have registered yourself for the CAT 2015 booster workshop series. The first workshop on Numbers will be on the coming Saturday.