Two particles A and B move with constant velocities v → 1 and v → 2 . At the initial moment their position…

Two particles A and B move with constant velocities v→1 and v→2. At the initial moment their position vectors are r→1 and r→2 respectively. The condition for particles A and B for their collision is:
  1. r→1⋅v→1=r→2⋅v→2
  2. r→1×v→1=r→2×v→2
  3. r→1-r→2=v→1-v→2
  4. r→1-r→2r→1-r→2=v→2-v→1v→2-v→1

Solution

For two particles to collide, the direction of the relative velocity of one with respect to other should be directed towards the relative position of the other particle
i.e. $\frac{\vec{r}_{1}-\vec{r}_{2}}{\left|\vec{r}_{1}-\vec{r}_{2}\right|}$ represents the direction of relative position of 1 w.r.t. 2. And $\frac{\vec{v}_{2}-\vec{v}_{1}}{\left|\vec{v}_{2}-\vec{v}_{1}\right|}$ represents the direction of velocity of 2 w.r.t. 1. So for collision of A and B, $\frac{\vec{r}_{1}-\vec{r}_{2}}{\left|\vec{r}_{1}-\vec{r}_{2}\right|}=\frac{\vec{v}_{2}-\vec{v}_{1}}{\left|\vec{v}_{2}-\vec{v}_{1}\right|}$.

Asked in: NEET 2015 (Phase 2)

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