Two particles A and B, move with constant velocities and . At the initial moment their position vectors are and respectively. The condition for particles A and B for their collision is [2015]

STUDY MATERIALS
COURSES
MORE
SUCCESS STORY
ADMISSION
STORE

Back

JEE ADVANCE

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

JEE MAIN

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

NEET

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

BITSAT

CLASS XI - XII

CLASS XI - XII

CBSE BOARD

CLASS IX

CLASS X

CLASS XI

CLASS XII

CLASS VI

CLASS VII

CLASS VIII

UP BOARD

CLASS IX
CLASS X
CLASS XI
CLASS XII

CLASS IX

CLASS X

CLASS XI

CLASS XII

CUET

CRASH COURSE

CRASH COURSE

Courses

CLASSROOM COURSES
ONLINE COURSES

ONLINE COURSES

More

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

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

Want Us to Email you About Special Offers & Updates?