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The 'e' Equation

Two point mas...

Question

Two point masses $1$ and $2$ move with uniform velocities $v_{1}$ and $v_{2}$ respectively. Their initial position vectors are $r_{1}$ and $r_{2}$ , respectively. Which of the following should be satisfied for the collision of the point masses?

A

\frac{r_{1} - r_{2}}{| r_{2} - r_{1} |} = \frac{v_{1} - v_{2}}{| v_{2} - v_{1} |}

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B

\frac{r_{2} - r_{1}}{| r_{1} - r_{1} |} = \frac{v_{2} - v_{1}}{| v_{2} - v_{1} |}

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C

\frac{r_{2} - r_{1}}{| r_{2} + r_{1} |} = \frac{v_{2} - v_{1}}{| v_{2} + v_{1} |}

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D

\frac{r_{2} + r_{1}}{| r_{2} + r_{1} |} = \frac{v_{2} - v_{1}}{| v_{2} + v_{1} |}

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Solution

The correct option is A $\frac{r_{1} - r_{2}}{| r_{2} - r_{1} |} = \frac{v_{1} - v_{2}}{| v_{2} - v_{1} |}$
For collision, $r_{1} + v_{1} t = r_{2} + v_{2} t$
$\Rightarrow r_{1} - r_{2} = (v_{2} - v_{1}) t$
Equating unit vectors, we get
$\frac{r_{1} - r_{2}}{| r_{2} - r_{1} |} = \frac{v_{1} - v_{2}}{| v_{2} - v_{1} |}$

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