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

Two particle A and B, move with constant velocities vec(v_1) and vec(v

1.	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
Answer» `vec(r_1) XX vec(v_1) = vec(r_2) xx vec(v_2)` `vec(r_1) - vec(r_2)= vec(v_1)- vec(v_2)` `(vec(r_1) - vec(r_2))/(\|vec(r_1) - vec(r_2)\|) = (vec(v_2) - vec(v_1))/(\|vec(v_2)- vec(v_1)\|)` `vec(r_1). vec(v_1) = vec(r_2) . vec(v_2)` Solution :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)\|)` .