Question

A particle starts from the origin of coordinates at time $t=0$ and moves in the $x-y$ plane with a constant acceleration α in the $y$ -diretion. Its equation of motion is $y= βx^{2}$ . Its velocity component in the $x$ -direction is

• A. Variable
• B. $\sqrt{\frac{2 \alpha }{\beta }}$
• C. $\frac{\alpha }{2 \beta }$
• D. $\sqrt{\frac{\alpha }{2 \beta }}$

Answer 1

Answer: (D)

Equation of motion is $y = \beta x^{2}$
$\frac{d^{2} y}{\textit{dt}^{2}} = \alpha $
$\frac{d^{2} x}{\textit{dt}^{2}} = 0$
from $y = x$
$\frac{\textit{dy}}{\textit{dt}} = 2 x \frac{dx}{\textit{dt}} \beta $
$\frac{d^2 y}{\mathrm{dt}^2}=\left(2\left(\frac{\mathrm{dx}}{\mathrm{dt}^2}\right)^2+2 \mathrm{x} \frac{d^2 x}{\mathrm{dt}^2}\right) \beta$
$a_{y} = \left(2 \left(\frac{\textit{dx}}{\textit{dt}}\right)^{2} + 0\right) \beta $
$a_{y} = 2 v_{x}^{2} \beta $
$v_{x} = \sqrt{\frac{a_{y}}{2 \beta }} = \sqrt{\frac{\alpha }{2 \beta }}$