The Question is from neet-physics , and it was created by the author SUBHANI

Question

Two particles A and B are moving with constant velocities $\vec{v_A}$ and $\vec{v_B}$ respectively. At $t=0$, their positions are $\vec{r_A}$ and $\vec{r_B}$.  The time at which their relative velocity is perpendicular to their relative position is:

SUBHANI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: Option B
## CONCEPT FRAMEWORK
    **Primary Concept**: Relative Motion, Dot Product
    **Related Topics**: Vector Algebra, Kinematics
    **Prerequisites**: Understanding of vectors, relative motion concepts, and dot product.
## SOLUTION PATHWAY
**Given Information**:
        Initial position of particle A: $\vec{r_A}$
        Initial position of particle B: $\vec{r_B}$
        Velocity of particle A: $\vec{v_A}$
        Velocity of particle B: $\vec{v_B}$
        Particles move with constant velocities.
**Method & Steps**:
    1. **Find relative position and relative velocity:**
        Reasoning: Relative position and velocity are essential for analyzing the condition.
        Formula/Rule: $\vec{r}_{rel}(t) = \vec{r}_B(t) - \vec{r}_A(t)$ and $\vec{v}_{rel} = \vec{v}_B - \vec{v}_A$
        Working: At time t, the position of particle A is $\vec{r}_A(t) = \vec{r}_A + \vec{v}_A t$ and the position of particle B is $\vec{r}_B(t) = \vec{r}_B + \vec{v}_B t$.  Thus, the relative position at time t is $\vec{r}_{rel}(t) = (\vec{r}_B + \vec{v}_B t) - (\vec{r}_A + \vec{v}_A t) = (\vec{r}_B - \vec{r}_A) + (\vec{v}_B - \vec{v}_A)t$.  The relative velocity is $\vec{v}_{rel} = \vec{v}_B - \vec{v}_A$.
    2. **Apply the perpendicularity condition:**
        Reasoning: Two vectors are perpendicular if their dot product is zero.
        Formula/Rule: $\vec{a} \cdot \vec{b} = 0$ if $\vec{a}$ and $\vec{b}$ are perpendicular.
        Working: We need to find the time t when $\vec{r}_{rel}(t) \cdot \vec{v}_{rel} = 0$.  Substituting the expressions from step 1, we get: $[(\vec{r}_B - \vec{r}_A) + (\vec{v}_B - \vec{v}_A)t] \cdot (\vec{v}_B - \vec{v}_A) = 0$. Expanding this dot product gives: $(\vec{r}_B - \vec{r}_A) \cdot (\vec{v}_B - \vec{v}_A) + t(\vec{v}_B - \vec{v}_A) \cdot (\vec{v}_B - \vec{v}_A) = 0$.
    3. **Solve for t:**
        Working:  $(\vec{r}_B - \vec{r}_A) \cdot (\vec{v}_B - \vec{v}_A) + t|\vec{v}_B - \vec{v}_A|^2 = 0$.  Therefore, $t = - \frac{(\vec{r}_B - \vec{r}_A) \cdot (\vec{v}_B - \vec{v}_A)}{|\vec{v}_B - \vec{v}_A|^2}$.  Since time cannot be negative, we consider the magnitude of the dot product.
        Result: $t = \frac{(\vec{r}_B - \vec{r}_A) \cdot (\vec{v}_B - \vec{v}_A)}{|\vec{v}_B - \vec{v}_A|^2}$ 
        Verification: The derived expression matches option B.
## OPTION ANALYSIS
[A] :x: This option represents the magnitude of relative position divided by magnitude of relative velocity, which does not represent the time when relative position and relative velocity are perpendicular.
[B] ✓: This option correctly represents the time at which the relative position and relative velocity are perpendicular, derived using the dot product condition for perpendicularity.
[C] :x: This option involves the sum of position vectors and velocities, which is incorrect in this context. The question is about relative motion, not the sum of individual motions.
[D] :x: This would mean the relative position and relative velocity are perpendicular at t=0, which is not generally true.
## LEARNING AIDS
    **Common Mistakes**:
        1. **Confusing magnitudes with dot products**: Using $\frac{|\vec{r}_B - \vec{r}_A|}{|\vec{v}_B - \vec{v}_A|}$ instead of the dot product condition.
        2. **Incorrect relative position**: Forgetting to consider the time component in the relative position at time 't'.
    **Quick Tips**:
        1. **Perpendicularity and Dot Product**: Two vectors are perpendicular when their dot product is zero, a useful tool in such problems.
        2. **Relative motion framework**: Always think of relative positions and velocities when dealing with motion of multiple objects.

SUBHANI · Answer

$\frac{|\vec{r_B} - \vec{r_A}|}{|\vec{v_B} - \vec{v_A}|}$

SUBHANI · Answer

The correct Option is "$\frac{(\vec{r_B} - \vec{r_A})\cdot(\vec{v_B} - \vec{v_A})}{|\vec{v_B} - \vec{v_A}|^2}$"

SUBHANI · Answer

$\frac{|\vec{r_B} + \vec{r_A}|}{|\vec{v_B} + \vec{v_A}|}$

SUBHANI · Answer

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