Velocity v of a moving particle on X-axis varies with x-coordinate varies as $v = β x^{1 / 3}$ , where $β$ is a positive constant. Assume that the particle starts from origin.

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Acceleration of particle is variable

Mean velocity of particle averaged over time T from starting is ${(\frac{2 β}{3})}^{3 / 2} \sqrt{T}$

Mean velocity of particle averaged over the time taken by the particle to move from Origin to the point, where its x-coordinate becomes $X_{0}$ is $\frac{2 β X_{0}^{1 / 3}}{3}$

All above statements are false

answer is A, B, C.

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Detailed Solution

$u = β x^{\frac{1}{3}} \frac{d x}{d t} = β x^{1 / 3} \int_{0}^{x} x^{- 1 / 3} d x = \int_{0}^{t} β d t \frac{3}{2} x^{2 / 3} = β t$

Mean Velocity = $\frac{x}{T}$ $= \frac{{(\frac{2 β T}{3})}^{3 / 2}}{T} = {(\frac{2 β}{3})}^{3 / 2} \sqrt{T}$ $a = v \frac{d v}{d x}$ $= β x^{1 /_{3}} (\frac{1}{3} β x^{- 2 / 3})$