Two point masses 1 and 2 move with uniform velocities $\mathbf{v}_1$ and $\mathbf{v}_2$ respectively. Their initial position vectors are $\mathbf{r}_1$ and $\mathbf{r}_2$, respectively. Which of the following should be satisfied for the collision of the point masses?

• Correct answer
  A

  $\frac{r_1-r_2}{\left|r_2-r_1\right|}=\frac{v_2-v_1}{\left|v_2-v_1\right|}$
• B

  $\frac{r_2-r_1}{\left|r_1-r_1\right|}=\frac{v_2-v_1}{\left|v_2-v_1\right|}$
• C

  $\frac{r_2-r_1}{\left|r_2+r_1\right|}=\frac{v_2-v_1}{\left|v_2+v_1\right|}$
• D

  $\frac{r_2+r_1}{\left|r_2+r_1\right|}=\frac{v_2-v_1}{\left|v_2+v_1\right|}$

Solution

$\begin{array}{ll}\text { For collision, } & \mathbf{r}_1+\mathbf{v}_1 t=\mathbf{r}_2+\mathbf{v}_2 t \\ \Rightarrow & \mathbf{r}_1-\mathbf{r}_2=\left(\mathbf{v}_2-\mathbf{v}_1\right) t\end{array}$
Equating unit vectors, we get $\frac{r_1-r_2}{\left|r_2-r_1\right|}=\frac{v_1-v_2}{\left|v_2-v_1\right|}$

Hence, the correct answer is option: 'A'.