Question

A particle projected from origin moves in $x-y$ plane with a velocity $\overrightarrow{v}=3\hat{i}+6x\hat{j}$, where $\hat{i}$ and $\hat{j}$ are the unit vectors 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. $y=2x^2$
• D. $y=\frac{1}{x}$

Answer 1

The correct option is A $y=x^2$

$V=3i+6xj$

$\begin{array}{c}V=\frac{dx}{dt}i+\frac{dy}{dt}j\\ \frac{dx}{dt}=\mathrm{3........}\left(1\right)\\ \frac{dy}{dt}=6x........\left(2\right)\\ fromequation\left(1\right)\\ dx=sdt\\ integratebothside\\ x=3t......\left(3\right)\\ takeequation\left(2\right)\\ dy=6xdt\\ puttingthevalueofxfrom3\\ dy=6\left(st\right)dt\\ dy=18tdt\\ integratebothside\\ y=9t^2\\ y=9{\left(\frac{x}{3}\right)}^2.......fromequation3\\ weget\\ y=x^2\end{array}$