Linear and Circular Races

Linear and circular races

                    

Commonly used terms

The terms given below are commonly used in this topic, and a clear understanding will help the student get a good grip on the subject.

Race: A contest of speed between participants is called a race.

Starting Point: The point from where a race begins is called the starting point.

Race Course: It means the path or ground on which races are run.

Finishing Point: The point where the race finishes is called the winning post or finishing point or a goal.

Dead Heat Race: A race in which no one is the winner because all the runners reach the winning post at the same time, is called a dead heat race.

Winner: The person who first reaches the finishing point is called the winner.

Winner's Time: The time taken by the winner to complete the race is called winner's time.

If A and B both start from the same place, then winner's distance = Length of the race.

Loser's Time: The time taken by the loser to complete the race ^y is called loser's time.

Linear Races

Suppose A & B are two contestants in a race. If before the start of the race, A is at the starting point and B is ahead of Q by 10 metres, then A is said to have given B a start of 10 metres. To cover a race of 300 metres in this case, A will have to cover a distance of 300 metres and B will have to cover (300 – 10) = 290 metres only.

Suppose A & B are two contestants in a race. At the end of the race, if A is at the finishing point and B is x metres away from the finishing point, then A is said to have beaten B by x metres in a race.

> A gives B a start of x metres implies that, if the distance between the starting point and finishing point is L metres, A covers L metres while B covers L – x metres.

 Starting point P                 R                                        Q   finishing point 

                                            X                  L – x      

From the figure given above, it Is clear that A starts at point P, but B starts at R at the same moment. For example in a 100 metres race, A gives B a start of 10 metres means, while A runs 100 metres, B runs 90 metres.

A beats B by x metres implies that, if the distance between the starting point and finishing point is L metres, A wins the race by covering L metres while B covers L – x metres only.

 Starting point P                                        R                   Q   finishing point 

                                                      L – x                     X

A gives B a start oft seconds, implies that A starts the race t seconds after B starts from the starting point.

A beats B by t seconds, implies that, A and B start together from the starting point, but A reaches the finishing point t seconds before B reaches.

Start distance is the distance between the two contestants at start if they are not starting from the same position.

If A & B are starting from the same point, A beats B by 'x' metres or 't' seconds means, B runs 'x' metres in 't' seconds.

El. In a 1760 m race, A gives B a start of 55 m and still beats him by 15 sec. If A runs @ 14.08 kmph, then find B's speed ?

Sol. While A runs 1760 m, B runs 1705 m.

Time taken by A to run 1760 m = 1760 x 60 /14080 = 7.5 min A beats B by 15 sec, hence time taken by B = 7.75 min. Now in 7.75 min, B runs 1705 m.

Hence speed = 1705/7.75 m / min = 13.2 kmph.

Beat distance is the distance between the winner and the loser when the winner reaches the finishing line.

Start time is the time difference between the time of start between the two competitors if they are not starting together.

Loser's distance = Winner's distance - (beat distance + start distance)

      Winner's time = Loser's time - (beat time + start time)

E2. In a 100 metres race, A runs at a speed of 2 m/sec. If A gives B a start of 4 metres and still beats him by 10 seconds, then find the speed of B.

Hence, A's time over the course = (175-7) = 168 seconds.

       Here A is the winner and B is the loser.

        Hence, Loser's time - Winner's time = beat time + start time

         B's time - A's time = 10 + 0.

         = 100 - 4       -  100    =   10  =  B's speed  = 1.6 m/sec.

            B speed          2

  

         Hence the speed of B is 1.6 m/sec.

Winner's time    =    Loser's time

Loser's distance        Winner's distance

   =  Beat time + Start time
       Beat distance + Start distance

E3  In a km race, A beats B by 40 metres or 7 seconds. Find A's time over the course.

     I. Here B runs 40 metres in 7 seconds.

B runs 1000 m in= 1000 x 7= 175 seconds

                                      40

Hence , A time over the course =  (175 – 7) = 168 seconds.

Short-cut: By formula,

Winner's time      =   Beat time + Start time

Loser's distance       Beat distance + Start distance

A's time = 7/ 40x 960 = 168 seconds.

If a race ends in a dead heat, then beat time = 0 and beat distance = 0.

Races of Three Participants

Suppose, A, B, and C participate in a race. The length of race is L metres. Assuming, A as the winner, i.e. A gets the first position. In the race, if A beats B by x metres, and A beats C by y metres, then values of x and y will decide the II position in the race.

If x < y, then B will beat C, i.e. B gets the II position.

If x > y, then C will beat B, i.e. C gets the II position.

The following relation is used for three participants in a race of same length

"' (L — x₁₂) x₂₃ = L (x₁₃ — x₁₂)

Where, Length of race = L. The first beats the second by a distance = x₁₂. The first beats the third by a distance = x₁₃. The second beats the third by a distance = x₂₃.

But, if the length of race (L) changes in each case, i.e.,
I beats II by x₁₂ metres in a race of L₁ metres.

I beats III by x₁₃ metres in a race of L₃ metres.

II beats III by x₂₃ metres in a race of L₂ metres.

So , x_12, x_{13 and} x₂₃  are to be comverted on a desire length of race , say , L metres.

Therefore x_{12  =}  x_{12  x  L    ,}  x₁₃  =  x₁₃  x  L    ,    x₂₃    =   x₂₃    x     L

                               L₁                        L₃                                     L₂

E4. In a race of 600 metres, A can beat B by 60 metres and in a race of 500 metres, B can beat C by 25 metres: By how many metres will A beat C in a 400 metres race?

Sol. Here, length of race is different in each race. So, respective beat distance (given) is to be converted to the desired length of race L(400 metres).

A is the winner (I).

Since B can beat C, therefore B becomes II and C becomes III in the race.

x₁₂= x₁₂ x L         =           60 x 400 = 40 metres.

             L₁                             600

X₂₃       =        X₂₃ x  L     =     25 x 400    =      20 metres, x₁₃ = ?

                                  L₂                     500

(L — x₁₂)x₂₃ = L(x₁₃ — x₁₂)

= (400 — 40) x 20 = 400(x₁₃ — 40)     

   -

X13=360 x 20  +  40  =  x₁₃=58metres.

           400

Hence A will beat C by 58 metres in a 400 metres race