Two particles A and B, move with constant velocities ${\vec{v}}_{1}$ and ${\vec{v}}_{2}$ . At the initial moment their position vectors are ${\vec{r}}_{1}$ and ${\vec{r}}_{2}$ respectively. The condition for particles A and B for their collision is [2015]

Question

Smart Achievers · Accepted Answer

(3)

Let the particles A and B collide at time $t$ . For their collision, the position vectors of both particles should be the same at time $t$ , i.e.,

${\vec{r}}_{1} + {\vec{v}}_{1} t = {\vec{r}}_{2} + {\vec{v}}_{2} t$ $;$ ${\vec{r}}_{1} - {\vec{r}}_{2} = {\vec{v}}_{2} t - {\vec{v}}_{1} t = ({\vec{v}}_{2} - {\vec{v}}_{1}) t$ ...(i)

Also, $| {\vec{r}}_{1} - {\vec{r}}_{2} |$ $=$ $| {\vec{v}}_{2} - {\vec{v}}_{1} |$ $t or t = \frac{| {\vec{r}}_{1} - {\vec{r}}_{2} |}{| {\vec{v}}_{2} - {\vec{v}}_{1} |}$

Substituting this value of $t$ in equation (i), we get

${\vec{r}}_{1} - {\vec{r}}_{2} = ({\vec{v}}_{2} - {\vec{v}}_{1}) \frac{| {\vec{r}}_{1} - {\vec{r}}_{2} |}{| {\vec{v}}_{2} - {\vec{v}}_{1} |}$ or $\frac{{\vec{r}}_{1} - {\vec{r}}_{2}}{| {\vec{r}}_{1} - {\vec{r}}_{2} |} = \frac{({\vec{v}}_{2} - {\vec{v}}_{1})}{| {\vec{v}}_{2} - {\vec{v}}_{1} |}$

Solution

Q.
Two particles A and B, move with constant velocities ${\vec{v}}_{1}$ and ${\vec{v}}_{2}$ . At the initial moment their position vectors are ${\vec{r}}_{1}$ and ${\vec{r}}_{2}$ respectively. The condition for particles A and B for their collision is [2015]

1 ${\vec{r}}_{1} \times {\vec{v}}_{1} = {\vec{r}}_{2} \times {\vec{v}}_{2}$

2 ${\vec{r}}_{1} - {\vec{r}}_{2} = {\vec{v}}_{1} - {\vec{v}}_{2}$

3 $\frac{{\vec{r}}_{1} - {\vec{r}}_{2}}{| {\vec{r}}_{1} - {\vec{r}}_{2} |} = \frac{{\vec{v}}_{2} - {\vec{v}}_{1}}{| {\vec{v}}_{2} - {\vec{v}}_{1} |}$

4 ${\vec{r}}_{1} \cdot {\vec{v}}_{1} = {\vec{r}}_{2} \cdot {\vec{v}}_{2}$

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Solution

Q. 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 [2015]

1 r→1×v→1=r→2×v→2

2 r→1-r→2=v→1-v→2

3 r→1-r→2|r→1-r→2|=v→2-v→1|v→2-v→1|