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?

Options

A. `vec(r_(1))xxvec(upsilon_(1))=vec(r_(2))xxvec(upsilon_(2))`
B. `(vec(r_(1))-vec(r_(2)))/(|vec(r_(1))-vec(r_(2))|)=(vec(upsilon_(2))-vec(upsilon_(1)))/(|vec(upsilon_(2))-vec(upsilon_(1))|)`
C. `vec(r_(1)).vec(upsilon_(1))=vec(r_(2)).vec(upsilon_(2))`
D. `vec(r_(1))xxvec(upsilon_(2))=vec(r_(2))xxvec(upsilon_(1))`

Answer

B

Explanation

Two particles `A` and `B` will collide if the direction of relative velocity of particle `B` with respect to particle `A[=(vec(upsilon_(2))-vec(upsilon_(1)))//|vec(upsilon_(2))-vec(upsilon_(1))|]` is directed towards the direction of relative position of partcle `A` with respect to particle `B[=(vec(r_(1))-vec(r_(2)))//|vec(r_(1))-vec(r_(2))|]` . Therefore, for collision of `A` and `B`
`(vec(r_(1))-vec(r_(2)))/(|vec(r_(1))-vec(r_(2))|)=(vec(upsilon_(2))-vec(upsilon_(1)))/(|vec(upsilon_(2))-vec(upsilon_(1))|)`