Well, this is quite a race we have here! So, A and B are running in opposite directions and they pass each other for the first time in 2 minutes. Let's figure out how long it takes for them to meet up again.
Since A is running at a constant speed v, and B is running at 2v, we can say that A covers v distance in 1 minute, and B covers 2v distance in 1 minute.
When they meet for the first time, let's assume they have covered a certain distance d. So, A has covered d/v minutes and B has covered d/(2v) minutes.
Since they take 2 minutes to meet for the first time, we can write an equation: d/v + d/(2v) = 2
Now, let's solve this equation to find the value of d.
Multiplying both sides of the equation by 2v, we get: 2d + d = 4v
Simplifying, we find: d = 4v/3
This means that they meet for the first time when they have covered (4v/3) distance.
Now, we want to know how much later they will pass each other for the second time. Since they meet for the first time after 2 minutes, we need to find the time it takes for them to cover an additional (4v/3) distance.
Since A covers distance v in 1 minute, and B covers distance 2v in 1 minute, we can say that A covers (4v/3) distance in (4v/3) minutes, and B covers (4v/3) distance in (2(4v/3)) minutes.
Adding these times, we get: (4v/3) + (8v/3) = 12v/3 = 4v
So, they will pass each other for the second time 4v minutes later.
And remember, while you're waiting for them to meet again, make sure to grab some popcorn and enjoy the show!