JEE PYQ: Motion in a Plane - Question ID 382b55b3f2d8 (JEE Main 2009)

In two-dimensional motion, a particle’s velocity and acceleration are vector quantities. The key formulas we use are:

• Velocity as a function of time: $\vec{v}(t) = \vec{v}_0 + \vec{a} \cdot t$ where $\vec{v}_0$ is the initial velocity vector, $\vec{a}$ is the constant acceleration vector, and $t$ is the time.
• Speed: The speed at any instant is the magnitude of the velocity vector: $v(t) = \sqrt{v_x^2(t) + v_y^2(t)}$ where $v_x(t)$ and $v_y(t)$ are the x- and y-components of the velocity at time $t$.

Since the acceleration is constant, we can directly integrate it to find the change in velocity.

Given:

• Initial velocity: $\vec{v}_0 = 3\widehat{i} + 4\widehat{j}$
• Acceleration: $\vec{a} = 0.4\widehat{i} + 0.3\widehat{j}$
• Time: $t = 10$ s

Step 1: Compute the velocity at $t = 10$ s. Using the formula $\vec{v}(t) = \vec{v}_0 + \vec{a} \cdot t$:

Simplify the components:

• $x$-component: $v_x(10) = 3 + 0.4 \cdot 10 = 3 + 4 = 7$
• $y$-component: $v_y(10) = 4 + 0.3 \cdot 10 = 4 + 3 = 7$

Thus, the velocity at $t = 10$ s is: $\vec{v}(10) = 7\widehat{i} + 7\widehat{j}$

Step 2: Compute the speed at $t = 10$ s. The speed is the magnitude of $\vec{v}(10)$: $v(10) = \sqrt{v_x^2(10) + v_y^2(10)} = \sqrt{7^2 + 7^2} = \sqrt{49 + 49} = \sqrt{98} = 7\sqrt{2}$

Conclusion: The speed after 10 s is $7\sqrt{2}$ units, which corresponds to option A.

1. Confusing speed with velocity: Some students mistakenly think the question asks for the magnitude of acceleration or the velocity vector itself. Always read carefully: the question asks for speed, which is a scalar (magnitude of velocity).

2. Incorrect component calculation: A frequent error is misapplying the acceleration to the velocity components. For example, adding $0.4 \cdot 10$ to the $y$-component or vice versa. Always double-check which component the acceleration affects.

3. Forgetting to take the square root: Some students compute $v_x^2 + v_y^2$ but forget to take the square root to find the magnitude. This leads to incorrect values like 98 instead of $7\sqrt{2}$.

Exam Tip: When dealing with vector components, always write them explicitly (e.g., $v_x$ and $v_y$) to avoid mixing them up. Also, verify units and dimensions at each step to catch calculation errors early.