Question

Two point masses 1 and 2 move with uniform velocities `vec(v)_(1)` and `vec(v)_(2)`, respectively. Their initial position vectors are `vec(r )_(1)` and `vec(r )_(2)`, respectively. Which of the following should be satisfied for the collision of the point masses?
A. `vecr_(1) - vecr_(2) = vecv_(1) - vecv_(2)`
B. `(vecr_(1) - vecr_(2))/(|vecr_(1) - vecr_(2)|) = (vecv_(2) - vecv_(1))/(|vecv_(2) - vecv_(1)|)`
C. `vecr_(1) . vecv_(2) = vecr_(2) . vecv_(2)`
D. `vecr_(1) xx vecv_(1) = vecr_(2) xx vecv_(2)`

Answer 1

For two partical to collides, the direction of the relative velocity of one with respect to other should be directed towards the relative position of the other partical.
The direction of relative position of `1`w.r.t. `2` is equal to unit vector of `vecr_(1.2). hatr_(1.2) = (vecr_(1) - vecr_(2))/(|vecr_(1) - vecr_(2)|)`
The direction of velocity of `2` w.r.t. `1`.
`hatr_(1.2) = (vecv_(2) - vecv_(1))/(|vecv_(2) - vecv_(1)|)`
For collision of `A and B` . `(vecr_(1) - vecr_(2))/(|vecr_(1) - vecr_(2)|) = (vecv_(2) - vecv_(1))/(|vecv_(2) - vecv_(1)|)`