Question

A particle is moving with velocity $v=k\left(y \hat{\text{i}} + x \hat{\text{j}}\right)$ where $k \, $ is a constant. The general equation for its path is

• A. $y=x^{2}+constant$
• B. $y^{2}=x+constant$
• C. $xy=constant$
• D. $y^{2}=x^{2 \, }+constant$

Answer 1

Answer: (D)

$v=k\left(y \hat{\text{i}} + x \hat{\text{j}}\right)$
$\frac{\text{d} \textit{x}}{\text{d} \textit{t}}=\textit{ky}$
$\frac{\text{dy}}{\text{d} \textit{t}}=\textit{kx}$
$\frac{\text{d} \textit{y}}{\text{d} \textit{x}}=\frac{\text{d} \textit{y}}{\text{d} \textit{t}}\cdot \frac{\text{d} \textit{t}}{\text{d} \textit{x}}=\frac{\left(\textit{kx}\right)}{\left(\textit{ky}\right)}$
$\left(\text{d} \textit{y}\right)\textit{ky}=\text{d}\textit{x.kx}$
$\textit{y}\text{dy}=\textit{x}\text{d}\textit{x}$
$\frac{\textit{y}^{2}}{2}=\frac{\textit{x}^{2}}{2}+\text{constant}$
$\Rightarrow \textit{y}^{2}=\textit{x}^{2}+\text{constant}$