Question

A particle projected from origin moves in $x-y$ plane with a velocity $\vec{v}=3 \vec{i}+6 x \hat{j}$, where $\vec{i}$ and $\hat{j}$ are the unit vector along $x$ and $y$ axis. Find the equation of path followed by the particle

• A. $y=x^{2}$
• B. $y=\frac{1}{x^{2}}$
• C. $y=2 x^{2}$
• D. $y=\frac{1}{x}$

Answer 1

Answer: (A)

Method 1: $\vec{V}=3 \hat{i}+6 x \hat{j}$
also $\vec{v}=\frac{d x}{d t} \hat{i}+\frac{d y}{d t} \hat{j}$
$\Rightarrow \frac{d x}{d t}=3$
$\int d x=\int 3 d t$
$x=3 t$
$\frac{d y}{d t}=6 x$
$d y=6 x \times d t$
$\int d y=\int 6 \times 3 t d t$
$=18 \int t d t \Rightarrow 18 \times \frac{t^{2}}{2}$
$y=9 t^{2}$
$=9 \times \frac{x^{2}}{9}$
$y=x^{2}$
Method 2:
$V_{x} \hat{i}+V_{y} \hat{j}=\vec{V}$
$V_{x}=3$
$V_{y}=6 x$
We know
$\frac{d_{y}}{d_{x}}=\tan \theta=\frac{V_{y}}{V_{x}}$
$\frac{d_{y}}{d_{x}}=\frac{6 x}{3 x}$
$\int_{0} d y=\int_{0} 2 x d x$
$y=x^{2}$