Two particle A and B, move with constant velocities `vec(v_1) and vec(v_2)`. At the initial moment their position vectors are `vec(r_1)` and `vec(r_2)` respectively . The condition for particles A and B for their collision is

Options

A. `vec(r_1) xx vec(v_1) = vec(r_2) xx vec(v_2)`
B. `vec(r_1) - vec(r_2) = vec(v_1) - vec(v_2)`
C. `(vec(r_1) - vec(r_2))/(|vec(r_1) - vec(r_2)|) = (vec(v_2) - vec(v_1))/(|vec(v_2) - vec(v_1)|)`
D. `vec(r_1). vec(v_1) = vec(r_2) . vec(v_2)`

Answer

C

Explanation

Let the particles A and B collide at time t. For their collision, the position vectors of both particles should be same at time t, i.e.
`vec(r_1) + vec(v_1) t = vec(r_2) + vec(v_2)t`
`vec(r_1) - vec(r_2) = vec(v_2) t = (vec(v_2) - vec(v_1)) t`
Also, `|vec(r_1) - vec(r_2)| = |vec(v_2) - vec(v_1)|t " or " t = (|vec(r_1) - vec(r_2)|)/(|vec(v_2) - vec(v_1)|)`
Substituting this value of t in eqn. (i), we get
`vec(r_1) - vec(r_2) = (vec(v_2) - vec(v_1)) (|vec(r_1) - vec(r_2)|)/(|vec(v_2) - vec(v_1)|) " or " (vec(r_1) - vec(r_2))/(|vec(r_1)-vec(r_2)|) = ((vec(v_2) - vec(v_1)))/(|vec(v_2) -vec(v_1)|)` .