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?
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?
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Two particles collide when their relative displacement is parallel (or anti-parallel) to their relative velocity; this ensures the separation is decreasing to zero.
Updated On: Apr 20, 2026
- \(\dfrac{\mathbf{r}_1 - \mathbf{r}_2}{|\mathbf{r}_1 - \mathbf{r}_2|} = \dfrac{\mathbf{v}_2 - \mathbf{v}_1}{|\mathbf{v}_2 - \mathbf{v}_1|}\)
- \(\dfrac{\mathbf{r}_2 - \mathbf{r}_1}{|\mathbf{r}_1 - \mathbf{r}_1|} = \dfrac{\mathbf{v}_2 - \mathbf{v}_1}{|\mathbf{v}_2 - \mathbf{v}_1|}\)
- \(\dfrac{\mathbf{r}_2 - \mathbf{r}_1}{|\mathbf{r}_2 - \mathbf{r}_1|} = \dfrac{\mathbf{v}_2 - \mathbf{v}_1}{|\mathbf{v}_2 + \mathbf{v}_1|}\)
- \(\dfrac{\mathbf{r}_2 + \mathbf{r}_1}{|\mathbf{r}_2 + \mathbf{r}_1|} = \dfrac{\mathbf{v}_2 - \mathbf{v}_1}{|\mathbf{v}_2 + \mathbf{v}_1|}\)
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept:
For collision: relative velocity must be along the line joining the two particles, i.e., along relative displacement.
Step 2: Detailed 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: Final Answer:
\[ \boxed{\dfrac{\mathbf{r}_1 - \mathbf{r}_2}{|\mathbf{r}_1 - \mathbf{r}_2|} = \dfrac{\mathbf{v}_2 - \mathbf{v}_1}{|\mathbf{v}_2 - \mathbf{v}_1|}} \]
For collision: relative velocity must be along the line joining the two particles, i.e., along relative displacement.
Step 2: Detailed 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: Final Answer:
\[ \boxed{\dfrac{\mathbf{r}_1 - \mathbf{r}_2}{|\mathbf{r}_1 - \mathbf{r}_2|} = \dfrac{\mathbf{v}_2 - \mathbf{v}_1}{|\mathbf{v}_2 - \mathbf{v}_1|}} \]
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