Q. A particle starts from the origin of coordinates at time $t = 0$ and moves in the $x - y$ plane with a constant acceleration $\alpha$ in the $y$-diretion. Its equation of motion is $y = \beta x^{2}$. Its velocity component in the $x$-direction is
A) Variable
B) $\sqrt{\frac{2\alpha}{\beta}}$
C) $\frac{\alpha}{2\beta}$
D) $\sqrt{\frac{\alpha}{2\beta}}$
Explanation
(d) $\frac{d^{2}y}{dt^{2}} = \alpha\ \ and\ \ \frac{d^{2}x}{dt^{2}} = 0$
$\frac{dy}{dt} = 2\beta x.\frac{dx}{dt}$
$\frac{d^{2}y}{dt^{2}} = 2\beta\left\lbrack x.\frac{d^{2}x}{dt^{2}} + \left( \frac{dx}{dt} \right)^{2} \right\rbrack$
$\because\ \alpha = 2\beta v_{x}^{2}$
$\therefore\ v_{x} = \sqrt{\frac{\alpha}{2\beta}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.