Having started from the same pint and at the same time, two runners A and B are running around a circular track of length 300m in opposite directions with speeds of 5m/s and 8m/s respectively. If they exchange their speeds after meeting for the first time, who will reach the starting point first?

Answer

To solve the problem step by step, let's break down the movements of runners A and B.

### Step 1: Calculate the time taken for A and B to meet for the first time.

**Formula**: When two objects move towards each other, their relative speed is the sum of their speeds.

- Speed of A = 5 m/s
- Speed of B = 8 m/s
- Relative speed = Speed of A + Speed of B = 5 m/s + 8 m/s = 13 m/s

**Distance**: Since they are running on a circular track of length 300 m, they will meet when they have covered this distance.

**Time to meet**:
\[ \text{Time} = \frac{\text{Distance}}{\text{Relative Speed}} = \frac{300 \text{ m}}{13 \text{ m/s}} \approx 23.08 \text{ seconds} \]

### Step 2: Calculate the distance each runner covers before they meet.

**Distance covered by A**:
\[ \text{Distance} = \text{Speed} \times \text{Time} = 5 \text{ m/s} \times 23.08 \text{ s} \approx 115.38 \text{ m} \]

**Distance covered by B**:
\[ \text{Distance} = \text{Speed} \times \text{Time} = 8 \text{ m/s} \times 23.08 \text{ s} \approx 184.62 \text{ m} \]

### Step 3: Determine the remaining distance each runner has to cover to reach the starting point after meeting.

**Remaining distance for A**:
- Total distance of the track = 300 m
- Distance covered by A = 115.38 m
\[ \text{Remaining distance for A} = 300 \text{ m} - 115.38 \text{ m} \approx 184.62 \text{ m} \]

**Remaining distance for B**:
- Total distance of the track = 300 m
- Distance covered by B = 184.62 m
\[ \text{Remaining distance for B} = 300 \text{ m} - 184.62 \text{ m} \approx 115.38 \text{ m} \]

### Step 4: Calculate the time taken for each runner to reach the starting point after they exchange speeds.

**New speeds after exchange**:
- Speed of A after exchange = 8 m/s
- Speed of B after exchange = 5 m/s

**Time taken by A to reach starting point**:
\[ \text{Time} = \frac{\text{Remaining Distance}}{\text{Speed}} = \frac{184.62 \text{ m}}{8 \text{ m/s}} \approx 23.08 \text{ seconds} \]

**Time taken by B to reach starting point**:
\[ \text{Time} = \frac{\text{Remaining Distance}}{\text{Speed}} = \frac{115.38 \text{ m}}{5 \text{ m/s}} \approx 23.08 \text{ seconds} \]

### Step 5: Compare the total time taken by both runners to reach the starting point.

- Total time for A = Time to meet + Time to reach starting point = 23.08 s + 23.08 s = 46.16 s
- Total time for B = Time to meet + Time to reach starting point = 23.08 s + 23.08 s = 46.16 s

### Conclusion:
Both runners A and B will reach the starting point at the same time.

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