A particle moving in the xy plane experiences a velocity dependent… | JEE Main 2020 (6 Sep, Shift 2)

Q27JEE Main 2020 · 6 SEP · SHIFT 2Difficult

Inertia, First & Second Laws

A particle moving in the xy plane experiences a velocity dependent force

$\vec{F} = k\left(v_y\hat{i} + v_x\hat{j}\right)$ , where vx and vy are the

x and y components of its velocity $\vec{v}$ . If $\vec{a}$ is the

acceleration of the particle, then

which of the following statements is true for the particle?

Show answer and solution

Answer: (b)

$\vec{a} = \frac{k}{m}(v_y\hat{i} + v_x\hat{j})$
$(\vec{v} \times \vec{a})_z = v_xa_y - v_ya_x = \frac{k}{m}(v_x^2 - v_y^2)$
$\frac{d}{dt}(v_x^2 - v_y^2) = 2v_xa_x - 2v_ya_y = \frac{2k}{m}(v_xv_y - v_yv_x) = 0$ , so $\vec{v} \times \vec{a}$ is constant.
(a), (c) fail: $\vec{v} \cdot \vec{a} = \frac{2k}{m}v_xv_y \neq 0$ and changes; (d) fails: a magnetic force is always $\perp \vec{v}$ .
So the answer is (b).

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