Alfred and Bernard run on the circular track of 600 m. Speeds of Alfred and Bernard are 30 m/s and 20 m/s respectively. Initially they are diametrically opposite to each other.
(i) When will they meet for the first time if both move in the same direction?
(ii) If both of them move in opposite directions, when will they meet for the second time?

Answer

To solve the problem step by step, we will break it down into two parts: (i) when Alfred and Bernard meet for the first time moving in the same direction, and (ii) when they meet for the second time moving in opposite directions.

### Part (i): Meeting in the Same Direction

1. **Identify the initial positions and speeds**:
- The circular track length is 600 m.
- Alfred's speed = 30 m/s.
- Bernard's speed = 20 m/s.
- Initially, they are diametrically opposite, which means they are 300 m apart.

2. **Calculate the relative speed**:
- Since they are moving in the same direction, the relative speed is the difference of their speeds.
- Relative speed = Alfred's speed - Bernard's speed = 30 m/s - 20 m/s = 10 m/s.

3. **Determine the time to meet**:
- They need to cover the initial distance of 300 m to meet.
- Time = Distance / Relative Speed = 300 m / 10 m/s = 30 seconds.

**Conclusion for Part (i)**: Alfred and Bernard will meet for the first time after **30 seconds**.

### Part (ii): Meeting in Opposite Directions

1. **Identify the initial positions and speeds**:
- The circular track length is still 600 m.
- Alfred's speed = 30 m/s.
- Bernard's speed = 20 m/s.
- Initially, they are still 300 m apart.

2. **Calculate the relative speed**:
- Since they are moving in opposite directions, the relative speed is the sum of their speeds.
- Relative speed = Alfred's speed + Bernard's speed = 30 m/s + 20 m/s = 50 m/s.

3. **Determine the time to meet for the first time**:
- They need to cover the initial distance of 300 m to meet.
- Time = Distance / Relative Speed = 300 m / 50 m/s = 6 seconds.

4. **Determine the time to meet for the second time**:
- After meeting for the first time, they will continue running. They will meet again when they cover the entire track length of 600 m.
- Time = Distance / Relative Speed = 600 m / 50 m/s = 12 seconds.

5. **Calculate the total time for the second meeting**:
- The total time until the second meeting is the time for the first meeting plus the time to cover the track.
- Total time = 6 seconds (first meeting) + 12 seconds (second meeting) = 18 seconds.

**Conclusion for Part (ii)**: Alfred and Bernard will meet for the second time after **18 seconds**.

### Summary of Answers:
- (i) They will meet for the first time after **30 seconds**.
- (ii) They will meet for the second time after **18 seconds**.