JEE PYQ: Motion in a Plane - Question ID 00437b0d3a73 (JEE Main 2019)

In problems involving relative motion in a plane, we treat the positions and velocities of the two moving objects as vectors. The key idea is:

• Express the position of Ship B relative to Ship A as a vector $\vec{r}(t)$ that changes with time.
• Express the velocity of Ship B relative to Ship A as a constant vector $\vec{v}_{\text{rel}}$.
• The distance between the two ships at any time $t$ is the magnitude of $\vec{r}(t)$. To find the time at which this distance is minimum, we minimize $|\vec{r}(t)|^2$ (which avoids dealing with square roots).

The fundamental formula is: $\vec{r}(t) = \vec{r}_0 + \vec{v}_{\text{rel}} \cdot t$ where $\vec{r}_0$ is the initial relative position vector and $\vec{v}_{\text{rel}}$ is the relative velocity vector.

1. Define the coordinate system and initial positions

• Let $\hat{i}$ point east and $\hat{j}$ point north.
• At $t=0$, Ship A is at the origin: $\vec{r}_A(0) = 0$.
• Ship B is initially at $\vec{r}_B(0) = 80\,\hat{i} + 150\,\hat{j}$ km relative to Ship A.

2. Write the velocity vectors

• Ship A’s velocity: $\vec{v}_A = 30\,\hat{i} + 50\,\hat{j}$ km/hr.
• Ship B’s velocity: $\vec{v}_B = -10\,\hat{i} + 0\,\hat{j}$ km/hr (since it moves west).
• Relative velocity of B with respect to A: $\vec{v}_{\text{rel}} = \vec{v}_B - \vec{v}_A = (-10 - 30)\,\hat{i} + (0 - 50)\,\hat{j} = -40\,\hat{i} - 50\,\hat{j}\;\text{km/hr}.$

3. Express the relative position vector at time $t$

4. Compute the squared distance $D^2(t)$

5. Expand and simplify

6. Minimize $D^2(t)$ by calculus

7. Match with the given options

1. Sign errors in relative velocity: Many students forget to subtract $\vec{v}_A$ from $\vec{v}_B$ or mix up the order. Always compute $\vec{v}_{\text{rel}} = \vec{v}_B - \vec{v}_A$. 2. Minimizing $D(t)$ instead of $D^2(t)$: Differentiating the square root is messier; minimizing the squared distance is algebraically cleaner. 3. Unit consistency: Ensure all distances are in km and velocities in km/hr so the time comes out in hours. 4. Approximation errors: Carry exact fractions (e.g., $107/41$) until the final step to avoid rounding mistakes.

Exam Tip: When the question asks for the time of minimum distance, always set up the relative position vector and minimize its squared magnitude. This method is robust and avoids trigonometric complications.