JEE PYQ: Motion in a Plane - Question ID fe40998c11cd (JEE Main 2010)

In problems involving motion in a plane, the velocity vector $\overrightarrow{v}$ is given as a function of position coordinates $(x, y)$. To find the path (trajectory) of the particle, we relate the components of velocity to the time derivatives of position:

• $\overrightarrow{v} = \frac{d\overrightarrow{r}}{dt} = v_x \widehat{i} + v_y \widehat{j}$
• $v_x = \frac{dx}{dt}$ and $v_y = \frac{dy}{dt}$

Given $\overrightarrow{v} = k(y \widehat{i} + x \widehat{j})$, we have:

• $v_x = \frac{dx}{dt} = ky$
• $v_y = \frac{dy}{dt} = kx$

To eliminate time $t$ and find a relation between $x$ and $y$, we use the chain rule:

This differential equation must be solved to obtain the path equation $y = f(x)$.

We start with the velocity components:

Divide equation (2) by equation (1) to eliminate $dt$:

This gives the differential equation:

Separate variables and integrate:

Integrate both sides:

Multiply both sides by 2 to simplify:

Let $2C = \text{constant} = C'$, so:

This matches option D: $y^2 = x^2 + \text{constant}$.

Students often make the following mistakes in this question:

1. Incorrect elimination of time: Some try to integrate $v_x$ and $v_y$ separately with respect to time without relating $x$ and $y$, leading to incorrect expressions.
2. Sign errors in integration: Forgetting the constant of integration or misplacing signs while integrating $y \, dy = x \, dx$.
3. Misidentifying the differential equation: Confusing $\frac{dy}{dx} = \frac{x}{y}$ with $\frac{dy}{dx} = \frac{y}{x}$ or other forms, leading to wrong path equations like $xy = \text{constant}$ (Option C).
4. Overcomplicating the problem: Attempting to solve for $x(t)$ and $y(t)$ explicitly, which is unnecessary and time-consuming.

Exam Tip: Always look for opportunities to eliminate time by dividing velocity components. This simplifies the problem to a first-order differential equation in $x$ and $y$, which is easier to solve.