# Circular Track YouTube Lecture Handout

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Three person A, B, C run along a circular path with speeds of 6 kmph, 4 kmph, 8 kmph respectively. If the length of circular path is 24 km. After what time will they meet again at starting point? Three person A, B, C run along a circular path with speeds of 6 kmph, 4 kmph, 8 kmph respectively.

## Circular Track

We can start important concepts you might already know this the concept of relative velocity so the relative velocity is the average rate with which the distance between two moving objects changes that՚s the relative velocity.

For example on a racetrack moves at the rate of 1 meter per second the second one moves at the rate of 2 meter per second in 1 second he would have come up to 2 meters this particular racer would have moved by then by 1 meter therefore the difference the distance between these two right these two racers after 1 second would be how much

So what is the average rate with which the distance these two is increasing it is 1 m/sec is the concept of the two objects are moving in same direction the rate with which the distance between these two changes is the difference between their individual rates.

The 2^{nd} case is when these two objects are moving towards each other these two racers are running towards each other what would happen then, this particular result is moving at the rate of 1 m/sec in this direction this particular one is moving at the rate of 2 m/sec in this direction so in 1 second what would have happened he would have come up to here 2 meters, he would have come up to here 1 meters so what is the rate at which the distance between these two changes or what is the relative velocity it is the addition of the individual rates it is the individual rates it is 3 m/sec

How the distance between these two is changing it is faster than both of their individual velocities moving on we can apply same concept in case of circular motion and that is what we would see now how to apply the same concept in case of circular motion.

There are two things to notice here there two racers will start and the distance between these two would grow grow grow and would they meet again that is another important concept in case of circular tracks when they meet again they would meet when the distance between these two has grown back to the circumference of the circle like the distance between these two increases to the circumference of circle they would meet again throughout the discussion we would take this circumference of the entire circle to be L that is our convention.

When the distance between these two increases – L these two racers would meet again on the circular track. So distance is growing growing growing it increase to L and then they meet again it increase to L right the center and they meet again right. So the relative velocity is same thing they are moving in the same direction so the distance is increasing if the difference between the individual velocities so that concept carries forward but the additional concept is when they cover the complete distance relative to each other they cover the distance of entire circumference of the circle it is L they meet again.

What happens in opposite direction so distance is increasing increasing increasing it increases to L and they made of this time the relative velocity is since they are travelling in opposite direction

Key Problems:

- Time when they first meet
- Time when they meet at starting point
- Number of times they meet (before one of them wins)

There are number of problems which are formed in the examination so, when do they first meet when do the first meet at this starting point this is meeting on anywhere on the circle when do they first meet on the starting point and number of times they made.

Variations:

- Direction of running
- Number of Participants
- Laps
- Head Start (Next Class)

This problem involving head start on circular tracks we would discuss in the next class.

Some sample problem and see how to solve such question again we will go by 1^{st} principles not getting distracted by shortcut tricks which really help.

## Three Person a and B Run Along a Circular Path with Speeds of 6 Kmph, 2 Kmph Respectively. If the Length of Circular Path is 24 Km. After What Time Will They Meet Again and Meet at the Starting Point Again? (Same Direction)

So there are two people A and B which are running on circular paths at the speed of 6 kmph and 2 kmph respectively.

Length of the path is 24, so the entire thing is 24 after what time will they meet again we are asked both the things and when they will meet at this starting point again so when will they first meet anywhere on the track and when will they meet at the starting point if they are travelling in the same direction.

So again what is the relative velocity: they are travelling in the same direction so relative velocity is the subtraction.

Then what is the length of the track kilometer

So, how much time would it take for them to cover a distance of 24 kilometer relative to each other so, what is the time

After 6 hours they would meet again. let՚s see what happens after six hours when they are meeting again.

After 6 hours, when they meet the first one would have traveled 36 kilometers the 2^{nd} would have traveled 12 kilometers.

The entire path is 24 km so we can that they would meet again here on this point this is the 1^{st} meet.

In the case of when they would first meet we have taken the relative velocity, 2^{nd} case when they will meet at the starting point we will not take relative velocity the relative velocity does not happen in this case.

Let us see how to find this, when they will meet at this starting point so when would this racer which travels 6 kmph reached the starting point he would reach:

2^{nd} racer would reach the starting point

When they happen to reach the starting point together then they would meet which means that the time would be a multiple of both 4 and 12 and we know how to find that is simply:

So they will meet at the starting point again after 12 hours

If you notice the racers would first meet bottom a certain amount of time double amount of time they would meet top.

## Three Person a and B Run Along a Circular Path with Speeds of 6 Kmph, 2 Kmph Respectively. If the Length of Circular Path is 24 Km. After What Time Will They Meet Again and Meet at the Starting Point Again? (Opposite Direction)

What happens in if they are travelling in opposite direction what is the relative velocity in opposite direction:

So again they will meet at the starting point after the same time the relative velocity does not factor there so that doesn՚t change in the opposite direction case that does not change what changes is the time when they meet again so let us calculate:

So entire 24 km they would cover now

So after 3 hours if they are travelling in opposite direction they would meet and again what are the distance they have covered is 6 km and 18 km.

This is the summary so far what we have done this is all that understood so far.

Time for meeting anywhere we have to use relative velocity we can calculate the relative velocity we know how to do that and then LCM of the relative time taken to length so they are more than two racers if there were two racers we compute the relative velocity and then we can compute the time at which they meet next.

If they are more than two racers then the LCM of their individual pairwise meeting time would give use the time when they will all meet together at any point on the circle.

When would we meet at the starting point a starting point calculate the time when each one reaches the starting point and then we take LCM of that time.

## Three Person a, B, C Run Along a Circular Path with Speeds of 6 Kmph, 4 Kmph, 8 Kmph Respectively. If the Length of Circular Path is 24 Km. After What Time Will They Meet Again at Starting Point?

Now three racers are running at the rate of 6 kmph, 4 kmph and 8 kmph. We asked when would they meet at the starting point they starting point they՚ll each reach the starting point after:

If you compute the LCM of these three we՚ll get the time when they all reach the starting point together so this one reaches every 4 hours this one which is starting every 3 hours and this one which is starting every 6 hours so they happen to be all together means that at that time should be a multiple of all these three which means it is 12.

LCM of

So, let us see when would they all meet they would meet when the pairwise they are meeting so if A and B meet, B and C meet, C and A meet then they meet all at the same time.

So, what is the relative velocity of A and B

B and C and A and C

The A and B would meet

Similarly, B and C meet and A and C meet

Take LCM of these three they would all meet at that time.

So, LCM of

So after 12 hours they will all be together and they will all be meeting at the same place.

## A Can Run One Full Round of a Circular Track in 6 Min and B in 15 Min. If Both a and B Start Simultaneously from the Same Starting Point then How Many Times Would They Meet in the Time B Has Completed 10 Rounds when Running in Same Direction, and in Opposite Direction?

This example to illustrate a point the point being that when we are computing relative velocity we can only add and subtract the speeds not the times that is the key.

So in this problem we have to compute the relative velocity because we are being asked when they would meet when how many times would they meet when B has completed 10 rounds.

So we have to compute the relative velocity when they are travelling in the same direction and they are travelling in opposite direction but we cannot add the time simply we have to complete the speed first so let assume that the length of the track

So,

In same direction,

Let take LCM of 6 and 15 is 30. Then

So, the relative velocity when they are travelling in same direction is .

So, after how many hours would they meet:

Time

So, after every 10 minutes they would meet when they travelling in the same direction.

It takes for meet to complete 10 rounds the speed is B completes 1 round in 15 minutes

So, 10 rounds will complete

Therefore in 150 minutes they would meet 15 times

What happen when they are travelling in opposite direction, we know the relative velocity would get added in that case.

So,

Time

They meet when they are travelling in opposite direction.

So,

Therefore, same direction and opposite direction

## In a 4000 Meter Race Around a Circumference of 1000 Meters, the Fastest and the Slowest Runner Reach the Same Point at the End of the 5^{th} Minute, for the First Time After the Start of 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?

The slowest racer travels at the speed

Let us then the fastest racer would travel at the speed

Then what is the relative velocity the track is 1000 meter.

Therefore, this is their relative velocity or

The faster racer is so that is

4000 meter race he would cover in how many minutes

## Number of Meeting Points

Another problem that asked the number of meeting point on a track so on a circular track. The two racers are running they are running with different speeds so obviously they will meet certain number of times and such kind of problems asks you to find number of times they would meet or positions of those point.

So let say how to solve such problems,

The length of the track

The relative speeds of two participants

Let us say for now that they are travelling in the same direction

Relative speed also assume that

So, relative speed

Distance that slower racer B,

If you look at this distance it is a fraction in b divided by so common factors in and , we can probably cancel them out let՚s say that they are all even number then we can remove two as a factor and then we can reduce this particular fraction to smaller terms:

The ration of the speed is .

We can notice that depends upon the ration of speed. This ration define when they meet again it means that this condition is exactly similar if I rotate this particular track in this direction this condition which had happened at the starting point so when would they meet next they would meet next at a distance of

When the two runners are running on a circular track the entire track can be divided into fractions which are equidistance from each other and the racers meet would meet these fractions not only are these their meeting point symmetrical distance they are not only equidistant they are also equally spaced in time.

let՚s take some examples:

There are two racers and I chosen different ratios for when they are meeting on the track so, the ratio is their speed so they would meet only one

Now, the ration is so they will be meeting twice

Again the entire track is divided into these many fraction those fraction are symmetrical in time and distance.

Now, this time the ration is so,

## Suppose a and B Are Running a 3 Km Race in a Circular Track of Length 300m. Speeds of a and B Are in the Ratio 4: 3. How Many Times and Where Would the Winner Pass the Other?

So ratio of the speed

How many times and where would the winner pass the other so, we are not asked the exact time when it would pass we are only asked how many times would pass it buy the time one of them wins the race probably

So, let՚s see how to solve this problem.

First we can assume, we are not asked absolute we know that the meeting points they only depend upon the ratio so as long as we keep the ration our answer should be correct.

So we take the minimum possible phase to simplify the problem which satisfy this requirement of

So we take these speeds to be 4 and 3 and solve the problem

it՚s a 3 km race but the length of the track

So, what would be the time at which they would meet they would meet after 300 seconds.

Then this person would have covered how much distance and

When the winner pass the other is starting point is 1.

They would meet **2 times**.

## X, Y and Z Move Along a Circular Path of Length 12 Km with Speeds of 6 Km/H, 8 Km/H and 9 Km/H Respectively. X and Y Move in the Same Direction but Z Moves in Opposite Direction. If They All Start at the Same Time and from Same Place, How Many Times Will X and Z Meet Anywhere on the Path by the Time X and Y Meet for the First Time Anywhere on the Path?

Now, X, Y and Z move along a circular path which is 12 km long.

X and Y move in the same direction but Z moves in opposite direction

So,

After 6 hours X and Y would meet at what time after how many time intervals X and Z meet

So,

So, **7 times** X and Z meet anywhere on the path by the time X and Y meet for the first time anywhere on the path.

## A and B Run in Opposite Directions on a Circle. A Runs in the Clockwise Direction. A Meets B First Time at a Point 500 M Away in Clockwise Direction. A Meets B Second Time at a Point 400 M Away in Anticlockwise Direction from Starting Point. If B is Yet to Complete One Round, What is the Circumference of the Circle?

So, there are two people A and B who are running in opposite directions on a circle and we are given two conditions.

So, the ratio of A and B:

So, the ratio of A and B

That the meeting points are symmetrical so, the 2^{nd} meeting point B has traveled 400 m

So, 1^{st} meeting point B

So,

## Next Time: Complex Problems in Circular Tracks (For E. G. With Head Start) , Clocks as Circular Tracks!

So the next class will solve some tricky problems which involved head starts which involved people is starting at different regions of the track in which involved which asked us to conditions when the participants reach different point in track not at one point but that symmetrically opposite to each other those kind of questions.

✍ Mayank