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# Time and Distance – Circular Races

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When two or more people travel in circular path , then it is trickier to find out their meeting point and meeting time.We would need mainly two things to conclude about various things about their meeting point and time. They are speed of each person,direction and length of the track. Sometimes instead of giving the total length of the path , radius of the circular path is given. In that case , we can always find out the total length of the path equal to circumference of the circle.

If radius of the circular path = r , the total length of path = $2\pi r$

Suppose the length of Circular path = L metre and speed of first person = m m/s and speed of second person = n m/s.
The meeting point of these two persons would also depend on whether they are moving in the same direction or opposite direction.

CASE I : When both are travelling in the same direction

Time when they would meet for the first time = $(\frac { L }{ m – n })$ sec

Time when they would meet for the first time at the origin = LCM of $(\frac { L }{ m } ,\frac { L }{ n } )$ sec

CASE I : When both are travelling in the opposite direction

Time when they would meet for the first time = $(\frac { L }{ m + n })$ sec

Time when they would meet for the first time at the origin = LCM of $(\frac { L }{ m } ,\frac { L }{ n } )$ sec

Suppose there are three people travelling with speed m , n and k m/s in the circular path of length L in the same direction, then

Time when all three would meet together for the first time = LCM of $(\frac { L }{ m – n },\frac { L }{ n – k })$ sec

Time when all three would meet for the first time together at the origin = LCM of $(\frac { L }{ m } ,\frac { L }{ n },\frac { L }{ k } )$ sec

Note : If the speed is given in km/hr , We can convert it into m/s by multiplying $\frac{5}{18}$ and If the speed is given in m/s , We can convert it into km/hr by multiplying $\frac{18}{5}$ .

M km/hr = $\frac{5M}{18}$ m/s

M m/s = $\frac{18M}{5}$ m/s

$\frac{5}{18}$ = $\frac{1000}{3600}$

$\frac{18}{5}$ = $\frac{3600}{1000}$