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 $y=\beta x^2$ . Its velocity component in the $x$ -direction is

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

Answer 1

Answer: (D)

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