Two particles, 1 and 2 moves with constant velocities v1 and v2. At the initial moment their radius vectors are equal to r1 and r2. How must these four vectors be interrelated for the particles to collide.

Solution:

Let the particles collide at the point A (Fig.) whose position vector is  \( {{\vec{r}}_{3}} \) (say). If t be the time taken by each particle to reach at point A, from triangle law of vector addition:

 \( {{\vec{r}}_{3}}={{\vec{r}}_{1}}+{{\vec{v}}_{1}}t={{\vec{r}}_{2}}+{{\vec{v}}_{2}}t \)

So,  \( {{\vec{r}}_{1}}-{{\vec{r}}_{2}}=({{\vec{v}}_{2}}-{{\vec{v}}_{1}})t\,\,\,\,\,\,\,\,\,\,(1) \)

Therefore,  \( t=\frac{\left| {{{\vec{r}}}_{1}}-{{{\vec{r}}}_{2}} \right|}{\left| {{{\vec{v}}}_{2}}-{{{\vec{v}}}_{1}} \right|}\,\,\,\,\,\,\,\,\,(2) \)

From Eqs. (1) and (2)

 \( {{\vec{r}}_{1}}={{\vec{r}}_{2}}-({{\vec{v}}_{2}}-{{\vec{v}}_{1}})\frac{\left| {{{\vec{r}}}_{1}}-{{{\vec{r}}}_{2}} \right|}{\left| {{{\vec{v}}}_{2}}-{{{\vec{v}}}_{1}} \right|} \)

Or,  \( \frac{{{{\vec{r}}}_{1}}-{{{\vec{r}}}_{2}}}{\left| {{{\vec{r}}}_{1}}-{{{\vec{r}}}_{2}} \right|}=\frac{{{{\vec{v}}}_{2}}-{{{\vec{v}}}_{1}}}{\left| {{{\vec{v}}}_{2}}-{{{\vec{v}}}_{1}} \right|} \), which is the sought relationship.

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