To solve the problem, let's break it down step by step.

Step 1: Understanding the Problem

• Pool Length: 50 meters
• Speeds: A = 5 m/s, B = 7 m/s
• Distance to be covered by each: 1000 meters
• They start from opposite ends, so they will be moving towards each other, cross, then continue back, repeating this until one of them completes 1000 meters.

Step 2: Time Taken to Meet

When A and B start from opposite ends, the relative speed at which they approach each other is the sum of their speeds: $\text{Relative Speed} = 5 \text{ m/s} + 7 \text{ m/s} = 12 \text{ m/s}$ The time it takes for them to meet for the first time: $\text{Time for First Meeting} = \frac{\text{Distance between them}}{\text{Relative Speed}} = \frac{50 \text{ m}}{12 \text{ m/s}} = \frac{25}{6} \text{ seconds} \approx 4.17 \text{ seconds}$ After this time, they will meet at a point 35 meters from A's starting point (and 15 meters from B's starting point).

Step 3: Distance Covered in Each Leg

Every time they meet, they have together covered 50 meters.

Step 4: Total Distance Covered to Win the Race

• Total distance for A: 1000 meters
• Total distance for B: 1000 meters

Step 5: Number of Meetings Before the Race Ends

Let's calculate the number of meetings by determining how far both A and B have traveled after each meeting until one of them reaches 1000 meters.

• Distance covered per meeting (both combined): 50 meters
• Number of meetings required to reach 1000 meters (per person): $\text{Number of Meetings} = \frac{1000 \text{ meters}}{50 \text{ meters}} = 20$

So they would cross each other 19 times before one of them completes 1000 meters because the 20th meeting will be after one of them has already finished the race.

Conclusion

A and B will meet 19 times before one of them wins the race.

Would you like any additional details or have any questions?

Here are some related questions for further exploration:

1. How long does it take for A to complete the race?
2. What is the distance between A and B when A finishes the race?
3. How many seconds after the 19th meeting does the race end?
4. If B were 1 m/s slower, how would that affect the number of meetings?
5. How far will B be from the finish line when A completes the race?
6. What is the total time taken for the entire race?
7. If the pool was 100 meters long, how many meetings would there be?
8. How would the number of meetings change if A's speed increased to 6 m/s?

Tip: When solving relative motion problems, always consider both the relative speed and the total distance to be covered.