Question

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?

• A. \(\dfrac{\mathbf{r}_1 - \mathbf{r}_2}{|\mathbf{r}_1 - \mathbf{r}_2|} = \dfrac{\mathbf{v}_2 - \mathbf{v}_1}{|\mathbf{v}_2 - \mathbf{v}_1|}\)
• B. \(\dfrac{\mathbf{r}_2 - \mathbf{r}_1}{|\mathbf{r}_1 - \mathbf{r}_1|} = \dfrac{\mathbf{v}_2 - \mathbf{v}_1}{|\mathbf{v}_2 - \mathbf{v}_1|}\)
• C. \(\dfrac{\mathbf{r}_2 - \mathbf{r}_1}{|\mathbf{r}_2 - \mathbf{r}_1|} = \dfrac{\mathbf{v}_2 - \mathbf{v}_1}{|\mathbf{v}_2 + \mathbf{v}_1|}\)
• D. \(\dfrac{\mathbf{r}_2 + \mathbf{r}_1}{|\mathbf{r}_2 + \mathbf{r}_1|} = \dfrac{\mathbf{v}_2 - \mathbf{v}_1}{|\mathbf{v}_2 + \mathbf{v}_1|}\)

Hint:

Two particles collide when their relative displacement is parallel (or anti-parallel) to their relative velocity; this ensures the separation is decreasing to zero.

Answer 1

The Correct Option is A

Step 1: Underlying Concept:
For collision: relative velocity must be along the line joining the two particles, i.e., along relative displacement. Step 2: Explanation:
At time \(t\): \(\mathbf{r}_1 + \mathbf{v}_1 t = \mathbf{r}_2 + \mathbf{v}_2 t\). Rearranging: \(\mathbf{r}_1 - \mathbf{r}_2 = (\mathbf{v}_2 - \mathbf{v}_1)t\). So displacement vector \((\mathbf{r}_1 - \mathbf{r}_2)\) must be parallel to relative velocity \((\mathbf{v}_2 - \mathbf{v}_1)\). This gives the unit vector condition (option A). Step 3: Conclusion:
\[ \boxed{\dfrac{\mathbf{r}_1 - \mathbf{r}_2}{|\mathbf{r}_1 - \mathbf{r}_2|} = \dfrac{\mathbf{v}_2 - \mathbf{v}_1}{|\mathbf{v}_2 - \mathbf{v}_1|}} \]