Circular Races: Theory and Concepts

Race is an act of running of two or more person to win or beat each other. The track or the path on which a race takes place is called a racecourse. The point from where a race starts is called the starting point, and the winning point or the finishing point is the point where a race finishes.
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The concept of Races can be further divided into two sections –
  1. Linear Race Track
    Racing on a linear track means running on a straight path such that the starting point and the finishing point don’t coincide with each other. Some general statements involved in such type of questions are as follows-
    • A gives a start of x metres to B implies if A starts the race from starting point, then B starts x metres ahead of A. To cover race of 100 metres in this case, A will have to cover 100 metres while B will have to cover only (100 - x) metres.
    • A beats B by x metres implies that in the same time, while A reached the winning point, whereas, B is behind A by x metres. Likewise, in a race of 100 metres in this case, A has covered 100 metres while B has covered only (100 - x) metres.
    • A can give B a start of t seconds implies that A will start t seconds after B starts from the starting point. Both A and B will reach the finishing point at the same time.
    • A beats B by x metres or t seconds implies that A and B start from the starting point at the same instant, but while A reaches the finishing point, B is behind by x metres, and, B takes t seconds more as compared to A to complete the race. So, B covers remaining x metres in extra t seconds. This gives the speed of B as x/t metres/second.
    • Dead Heat or Dead Lock is a situation when all participants reach the finishing point at the same instant of time.
  2. Circular Race Track
    Racing on a circular track means running on a circular path such that the starting point and the finishing point coincide with each other. Let two athletes U and V started running on a circular track of length L metres from a same starting point P. The speed of U is u m/s and that of V is v m/s. (Kindly take care of the units)
    The questions that may follow-
    1. When will they meet at starting point for the first time?
      Time taken to complete one round = Length of the circular track/speed
      ⇒ Time taken by U and V to complete one round is L/u and L/v respectively.
      Circular Races Theory and Concepts
      Time taken to meet for the 1st time at starting point, T = LCM (L/u, L/v)
      Even in case of more than two runners, lets’ say U, V and W with speeds u m/s, v m/s and w m/s respectively.
      Time taken to meet for the 1st time at starting point = LCM (L/u, L/v, L/v)
      For example, let u = 8 m/s, v = 5 m/s and w = 10 m/s and length of the circular track = 400m.
      Time taken to meet for the 1st time at starting point = LCM (400/8,400/5,400/10) = LCM(400,400,400)/HCF(8,5,10) = 400 sec
    1. When will they meet for the first time on the circular track? (not necessarily at the starting point)
      Time taken to meet for the 1st time = Length of the track/Relative Speed
      Case 1: While they are moving in opposite direction, time taken = L/u+v
      Case 2: While they are moving in same direction, time taken = L/u-v
      In case of more than two runners, lets’ say U, V and W with speeds u m/s, v m/s and w m/s respectively.
      Time taken to meet for the 1st time by U, V and W = LCM (Time taken to meet for the 1st time of any two out of three)
      For example, let u = 8 m/s, v = 5 m/s and w = 10 m/s and length of the circular track = 400 m. Also U and V are moving in same direction while W is moving in opposite direction.
      ⇒ Time taken by U and V to meet for the 1st time = 400/8-5 = 400/3 sec and time taken by V and W to meet for the 1st time= 400/5+10 = 400/15 sec.
      ⇒ Time taken by U, V and W to meet for the 1st time = LCM (400/3,400/15) = LCM(400,400)/HCF(3,15) = 400/3 sec
    2. At how many points will they meet while moving in opposite direction?

      Number of meeting points=
      Circular Races Theory and Concepts
      For example, let u = 8 m/s and v = 5 m/s and length of the circular track = 400 m. Also U and V are moving in opposite direction.
      Number of meeting points = Circular Races Theory and Concepts = 13 times
    3. At how many points will they meet while moving in same direction?
       
      Number of meeting points
      Circular Races Theory and Concepts
      For example, let u = 8 m/s and v = 5 m/s and length of the circular track = 400 m. Also U and V are moving in same direction.
      Circular Races Theory and Concepts
      Even in case of more than two runners, the process remains same.
      For example, let u = 8 m/s, v = 5 m/s and w = 10 m/s and length of the circular track = 400 m. Also U and V are moving in same direction while W is moving in opposite direction.
      Number of meeting points = Circular Races Theory and Concepts = 3 times {refer point (iii)}
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