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)

$\frac{ dv _{x}}{ dt }=\frac{ k }{ m } v _{ y }$
$\frac{ dv _{y}}{ dt }=\frac{ k }{ m } v _{ x }$
$\frac{ dv _{ y }}{ dv _{ x }}=\frac{ v _{ x }}{ v _{ y }} $
$\Rightarrow \int v _{ y } dv _{ y }=\int v _{ x } dv _{ x }$
$v _{ y }^{2}= v _{ x }^{2}+ C$
$v _{y}^{2}- v _{ x }^{2}$ = 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