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Q.

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

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a

$α / β$

b

$α / 2 β$

c

$\sqrt{2 α β}$

d

$\sqrt{α / 2 β}$

answer is D.

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Detailed Solution

$\begin{array}{l} \frac{d^{2} y}{{dt}^{2}} = α and \frac{d^{2} x}{{dt}^{2}} = 0 \\ y = {βx}^{2} \\ ∴ \frac{dy}{dt} = 2 βx \cdot \frac{dx}{dt} \\ \frac{d^{2} y}{{dt}^{2}} = 2 β [x \cdot \frac{d^{2} x}{{dt}^{2}} + {(\frac{dx}{dt})}^{2}] \\ ∴ α = 2 {βv}_{x}^{2} ∴ v_{x} = \sqrt{α / 2 β} \end{array}$

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