4 tortoises are at 4 corners of a square with side a. They being to move at the same time with a constant speed v , the first one ialwys moving in a direction towards the second & so on/ Find the time aftr which they meet ?

the motion will be that in the form of a spiral terminating at the centre of the square since they are symmetrical 

this diagram represents the initial scenario at  t=0 

so the velocity of A has two components i.e. cos 45 and sin 45 with the angle remaining constant throughout

now our job is to find the total distance it travels on reaching the centre O

when A moves d distance forward towards B it moves d cos45 closer to centre O or if it has to move 1 metre closer to O it has to move 1/(cos 45) metre or for 1metre closer to O it has to run 1.41 metre 

total it has to travel a cos45 to get to O and therefore he has to run 1/(cos 45) times this distance which is a

so the time taken for the dogs to meet will be  a/v which is the required answer.