Continuous moving buses puzzle

2 busses. One after 20 min and the other after 30 min. 3 if we can include the bus pulling out when this Bus reaches its destination.

5 buses including the one starting when the bus reach its destination.

In simple words the bus had to pass all the buses from opposite direction before its reach its destination.

So during 40 minute interval there are 5 buses from opposite direction.

bus1 which start at 0 minute which our bypasses at 20 minute

bus2 which starts at 10 minute which our bypasses at 30 minute

bus3 which starts at 20 minute which our bypasses at 35 minute

bus4 which starts at 30 minute which our bypasses at 37.5 minute

and bus5 which starts at 40 minute(at the destination) which our bus bypasses at 40 minute

None!!

Here t denotes time in minutes.
X denotes path not visited by any bus yet.
b1,b3,b5……..denotes the buses which start from left side of the road
while b2,b4,b6……denotes the buses which start right side of the road!!!
(number)…..denotes how many buses they meet on the way…….

when t=0;    b1(0)———X———-X———-X———b2(0)

when t=10;    b3(0)———b1(0)———-X———-b2(0)———b4(0)

when t=20;    b5(0)———b3(0)———-b1(1)b2(1)———-b4(0)———b6(0)

when t=30;    b7(0)———b5(1)b2(2)———-b3(1)b4(1)———-b1(2)b6(1)———b8(0)

when t=40; ……………………………………..every bus will meet at least one bus during its path from one end to another.

Answer – The total number of buses met will be 9

Explanation-
EXCEPT for the FIRST 3 buses starting from either end, every other bus will meet another bus from the opposite site every 5 minutes.  This is because buses from both ends are moving simultaneously (covering equal distances, assuming buses moving at uniform speed).

Let us denote the distance covered in every minute as a point. i.e, considering one end of the route as Point 0, the other end will be Point 40. Consider a bus (Say, A) starting from point 0. The moment it starts, another bus B which had started 30 minutes earlier from the other end (Point 40) will be  already at 10 min. away (Point 10) from reaching  its destination. By the time bus A covers the first 5 points of its journey, the bus B also will have covered the same distance (from Point 10) and hence will meet bus A. By that time, yet another bus C (started from Point 40 following bus B)will be at 10 points away, and this bus will meet A in the next 5 minutes, just like bus B did. Thus, bus A will meet buses B, C… etc. every five minutes. This will continue till it reaches Point 40. Thus, the number of buses it meets en route will be 7. Counting the buses it meets at the starting point (the bus which had left Point 40, at 40 min. earlier) and at the terminating point (the bus which leaves Point 40 at the 40th minute of Bus A’s journey), the total number of buses met will be 9.

(For the first 3 buses starting at either end, this calculation will not apply, because, when the first bus starts from Point 0, there is no bus which had started earlier from Point 40 on the way to meet it at Points 5, 10 and 15. It will meet the first bus from the opposite end at Point 20 only. Thereafter, it will meet subsequent buses as described above. Similarly, when the 2nd bus leaves Point 0, there will be one bus at Point 30 (which was the first bus from Point 40, left 10 min, earlier) which will be the first bus it meets en route at Point 15. followed by subsequent buses as above.)