Two boys A and B are at two diametrically opposite points on a circle. At one

instant the two start running on the circle; A anticlockwise with constant speed v
and B clockwise with constant speed 2v. In 2 minutes, they pass each other for
the first time. How much later will they pass each other for the second time?

distancetraveled=2V*t+V*t

PIr=3V*2
V=PIr/6

now for the second passing, the distance will be twice as far allthe way around, or 2PIr
distance traveled=v*t+2vt
2PI r=3Pir/6 * t
t=4 minutes

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!

To solve this problem, we need to understand the relative speeds of A and B and how they affect their meeting points on the circle.

Let's say the circumference of the circle is C. Since A is running anticlockwise with a constant speed v, he covers a distance of v in 1 minute. Similarly, B, running clockwise with a speed of 2v, covers a distance of 2v in 1 minute.

Now, we know that A and B started from diametrically opposite points. When they meet for the first time after 2 minutes, they have covered a total distance of half the circumference.

Distance covered by A in 2 minutes = 2v
Distance covered by B in 2 minutes = 4v

Since A and B meet at a point after running half the circumference, their total distance covered is C/2.

Therefore, we can set up the equation:
2v + 4v = C/2

Simplifying this equation, we get:
6v = C/2
v = C/12

So, A's speed is C/12 and B's speed is 2v = 2(C/12) = C/6.

Now, to find the time it takes for A and B to pass each other for the second time, we need to determine when their total distance covered is equal to the circumference of the circle (C).

Distance covered by A in t minutes = vt = (C/12)t
Distance covered by B in t minutes = 2vt = (C/6)t

Setting their total distance equal to C, we get:
(C/12)t + (C/6)t = C

Simplifying this equation, we get:
(t/12) + (2t/12) = 1
(3t/12) = 1
t = 4 minutes

Therefore, A and B will pass each other for the second time after 4 minutes.

In summary:
1. Calculate A's speed: v = C/12
2. Calculate B's speed: 2v = C/6
3. Set up the equation: (C/12)t + (C/6)t = C
4. Solve for t: t = 4 minutes

let the distance travel by both be a&b

a has a distance of d1 like wise b
speed=distanc/time
d1=2v
d2=4v
total distance
d1+d2=180
2v+4v=180
6v=180
v=30
d1=2*30
d1=60
d2=4*20
d2=120
now start u get u starting

not that there meet at same point so now calculate the time they will trave in there various direction
time=distance/velocity
but they but move at a standard velocity