Solved: PHYSICS Question for NEET

NEET PHYSICSMedium

Two particles $A$ and $B$ , move with constant velocities $\vec{v}_1$ and $\vec{v}_2$ respectively. 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 will be:

$\frac{\vec{r}_1 - \vec{r}_2}{|\vec{r}_1 - \vec{r}_2|} = \frac{\vec{v}_2 - \vec{v}_1}{|\vec{v}_2 - \vec{v}_1|}$

$\vec{r}_1 \cdot \vec{v}_1 = \vec{r}_2 \cdot \vec{v}_2$

$\vec{r}_1 \times \vec{v}_1 = \vec{r}_2 \times \vec{v}_2$

$\vec{r}_1 - \vec{r}_2 = \vec{v}_1 - \vec{v}_2$

Step-by-Step Solution

Condition for Collision: For two particles to collide, they must occupy the same position at the same instant of time $t$ ( $t > 0$ ).
Position Equations:

Position of particle A at time $t$ : $\vec{r}_A(t) = \vec{r}_1 + \vec{v}_1 t$
Position of particle B at time $t$ : $\vec{r}_B(t) = \vec{r}_2 + \vec{v}_2 t$

Equating Positions: $\vec{r}_1 + \vec{v}_1 t = \vec{r}_2 + \vec{v}_2 t$ $\vec{r}_1 - \vec{r}_2 = (\vec{v}_2 - \vec{v}_1) t$
Direction Analysis: The equation implies that the relative displacement vector $(\vec{r}_1 - \vec{r}_2)$ is parallel to the relative velocity vector $(\vec{v}_2 - \vec{v}_1)$ . Since time $t$ must be positive for a future collision, these two vectors must point in the same direction.
Unit Vectors: If two vectors are in the same direction, their unit vectors must be equal . $\hat{r}_{relative} = \hat{v}_{relative}$ $\frac{\vec{r}_1 - \vec{r}_2}{|\vec{r}_1 - \vec{r}_2|} = \frac{\vec{v}_2 - \vec{v}_1}{|\vec{v}_2 - \vec{v}_1|}$

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