Races and Games - Important Points
Race is a competition of speeds. All participants of a race are required to run a specific distance; whoever does it in the minimum time, is the winner of the race.
Linear Races
Terminology related to linear races:
Dead Heat: When all participants reach the finishing point at the same instant of time, the race is said to end in a "Dead Heat".
A beats B by x meters: This means, both A and B start at the starting point at the same instant of time, but while A reaches the finishing point, B is x meters behind. This indicates that A is the winner of the race. From this, we can get ratio of their speeds which is same as the ratio of distances covered by them.
A beats B by t seconds: This means, both A and B start at the starting point at the same instant, but B takes t seconds more as compared to A to finish the race. So, A is the winner and we can calculate speed of B if both the above terms are given.
A gives B a start of y meters: A starts from the starting point and B starts y meters ahead of A.
A can give B a start of y meters: A starts from the starting point and B starts y meters ahead of A, but still both A and B reach the finishing point at the same instant of time. So, the race ends in a dead heat.
A gives B a start of t seconds: A starts t seconds after B starts from the starting point.
A can give B a start of t seconds: A starts t seconds after B starts from the starting point, but still, both A and B reach the finishing point at the same instant of time. So, the race ends in a dead heat.
A gives B, y meters and t seconds: A and B start at the starting point at the same instant, but while A reaches the finishing point, B is behind by y meters, and, B takes t more seconds to complete the race. This gives the speed of B as y m /t secs.
Circular Races
The relative speeds of 2 bodies moving around a circle in the same direction is taken as (S1 - S2) and while moving in opposite direction is taken as (S1 + S2)
First Meeting: When three or more bodies start moving simultaneously from the same point on the circumference of the circle, they will 1st meet again in the time given by LCM of the times that the fastest runner takes in totally overtaking each of the slower runners.
First Meeting at starting point: If two or more bodies start moving simultaneously from the same point on the circumference of the circle, they will 1st meet again at the starting point in the time given by LCM of the time taken by individual to complete one round. They will continue to meet at the starting point at this interval.
- When running in same direction on a circular track, the faster one meets the slower one whenever he/she gains one full round over the slower one.
- If the faster one has gained n rounds over the slower one, they have met n times. Hence if faster one has run x rounds and slower one has run y rounds, they have met [x - y] times where [ ] refers to the integral part.
- When running in opposite direction, the will meet if ‘together’ they complete one round. Thus if faster one covers x rounds and in the same time slower one covers y rounds, they have met [x + y] times.