Q:

How to decide which velocity is bigger in a collision before writing velocity of approach and velocity of separation?

My teacher said that velocity of separation is $V_1$- $V_2$ where $V_1$ is greater than $V_2$ and said same for velocity of approach. The problem is how would you determine which velocity is greater if the question requires you to find an unknown velocity . We can make mistakes in writing correct equations.
Hope you got my point :)

A:

The trick here is to use relative velocity of approach= negative relative velocity of seperation. By defining a specific direction to be positive and then assigning a direction(sign) to every velocity, all u need to do is plug in the signed velocities into the vector equation
I have given the derivation of the formula below with velocities of completely arbitrarily direction ( the vector sign shows that the direction can be anything even in 3d space!). I must admit that I just added the vector signs in the k.e equation instead or doing it the right way using manipulation of the dot product, but that will just confuse u.
This shows that one u have chosen a specific direction to ge positive, just use the momentum and reltive velocity equations in vector form
I have included a solved example using vectors, which just take care of themselves without you having to worry about keeping anything  positive

