Understanding the Circular Race Problem

This problem involves a circular race where two individuals, a man and a woman, start from the same point and move in opposite directions. We are given the length of the circular track and their speeds. The goal is to find the time it takes for them to meet for the first time after the start.

Key Concept: Relative Speed in Opposite Directions

When two objects move towards each other or in opposite directions on a path (including a circular path), their speeds add up to determine how quickly the distance between them decreases. This combined speed is called their relative speed.

In a circular race, when two participants start from the same point and move in opposite directions, the distance they need to cover together to meet for the first time is equal to the length of the track.

Step-by-Step Calculation of Meeting Time

Let's break down the problem and calculate the time they meet for the first time.

1. Identify the given values:
   • Length of the circular track (Distance), $D = 2500 \text{ m}$.
   • Speed of the man, $S_m = 37 \text{ km/h}$.
   • Speed of the woman, $S_w = 35 \text{ km/h}$.
2. Convert speeds to a consistent unit:

   The distance is given in meters, but the speeds are in kilometers per hour. We should convert the speeds to meters per second (m/s) or convert the distance to kilometers.

   Let's convert speeds to m/s. The conversion factor is $\frac{5}{18}$ (since $1 \text{ km/h} = \frac{1000 \text{ m}}{3600 \text{ s}} = \frac{5}{18} \text{ m/s}$).

   • Man's speed in m/s: $S_m = 37 \times \frac{5}{18} \text{ m/s} = \frac{185}{18} \text{ m/s}$.
   • Woman's speed in m/s: $S_w = 35 \times \frac{5}{18} \text{ m/s} = \frac{175}{18} \text{ m/s}$.

   Alternatively, we can convert the track length to kilometers: $D = 2500 \text{ m} = 2.5 \text{ km}$. Then we can work with speeds in km/h and the answer in hours, finally converting to minutes and seconds.

   Let's use the km and km/h units first, as it might be simpler for relative speed calculation.

   • Track length, $D = 2.5 \text{ km}$.
   • Man's speed, $S_m = 37 \text{ km/h}$.
   • Woman's speed, $S_w = 35 \text{ km/h}$.
3. Calculate the relative speed:

   Since they are moving in opposite directions on the circular track, their relative speed is the sum of their individual speeds.

   Relative Speed, $S_{relative} = S_m + S_w = 37 \text{ km/h} + 35 \text{ km/h} = 72 \text{ km/h}$.

4. Calculate the time to meet for the first time:

   The time it takes for them to meet for the first time is the time required to cover the total distance of the track at their relative speed.

   Time, $T = \frac{\text{Distance}}{\text{Relative Speed}} = \frac{D}{S_{relative}}$.

   $T = \frac{2.5 \text{ km}}{72 \text{ km/h}} \text{ hours}$.

   $T = \frac{2.5}{72} \text{ hours}$.

5. Convert the time to minutes and seconds:

   The time is currently in hours. Let's convert it to seconds for clarity, as the options are in minutes and seconds. There are 3600 seconds in an hour.

   $T = \frac{2.5}{72} \times 3600 \text{ seconds}$.

   $T = \frac{2.5 \times 3600}{72} \text{ seconds}$.

   $T = \frac{9000}{72} \text{ seconds}$.

   Let's simplify the fraction:

   $T = \frac{4500}{36} = \frac{2250}{18} = \frac{1125}{9} = 125 \text{ seconds}$.

   Now, convert 125 seconds into minutes and seconds. 1 minute = 60 seconds.

   $125 \text{ seconds} = 2 \times 60 \text{ seconds} + 5 \text{ seconds} = 2 \text{ minutes and } 5 \text{ seconds}$.

Conclusion

The man and the woman will meet for the first time 2 minutes and 5 seconds after the start of the circular race.

Revision Table: Key Formulas

Additional Information on Circular Track Races

Circular track race problems often involve concepts of relative speed and Least Common Multiple (LCM) when dealing with multiple meetings or when participants run at different speeds in the same direction.

• Meeting for the first time (Same Direction): If two people start at the same point and run in the same direction on a circular track, they will meet for the first time when the faster person has gained one full lap (one track length) over the slower person. The time taken is $\frac{\text{Track Length}}{|S_1 - S_2|}$.
• Meeting at the starting point (Same or Opposite Direction): To find when they meet *at the starting point* for the first time, you need to calculate the time each person takes to complete one lap ($\frac{\text{Track Length}}{\text{Speed}}$ for each). The time they meet again at the starting point is the LCM of the individual lap times.
• Multiple Meeting Points: The number of distinct points where two people meet on a circular track during a race depends on their speeds and the track length.
• Units are Crucial: Always ensure all quantities (distance, speed, time) are in consistent units before performing calculations. Common units are meters and seconds (m/s) or kilometers and hours (km/h).

Understanding relative speed is fundamental to solving these types of problems efficiently.