Question

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 $v_x$ and $v_y$ 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 ?

• A. quantity $\vec{v}\cdot \vec{a}$ is constant in time
• B. kinetic energy of particle is constant in time
• C. quantity $\vec{v}\times \vec{a}$ is constant in lime
• D. $\vec{F}$ arises due to a magnetic field

Answer 1

Answer: (C)

$\frac{d{v}_x}{dt}=\frac{k}{m}{v}_y$
$\frac{d{v}_y}{dt}=\frac{k}{m}{v}_x$
$\frac{d{v}_y}{d{v}_x}=\frac{v_x}{v_y}$
$\Rightarrow \int v_{y}d{v}_y=\int v_{x}d{v}_x$
$v_y^2=v_x^2+C$
$v_y^2-v_x^2$ = constant
Option (3)
$\vec{v}\times \vec{a}=\left(v_x\hat{i}+v_y\hat{j}\right)\times \frac{k}{m}\left(v_y\hat{i}+v_x\hat{j}\right)$
$=\left(v_x^2\hat{k}-v_y^2\hat{k}\right)\frac{k}{m}$
$=\left(v_x^2-v_y^2\right)\frac{k}{m}\hat{k}$
= Constant