Question

Velocity $V$ of a moving point on the $x$-axis varies with its $x$ coordinate as $V=\beta x^{\frac{1}{3}}$ where $\beta$ is a positive constant. Assuming the particle to start from the origin

• A. Acceleration of particle is variable.
• B. Mean velocity of point averaged over time $T$ from starting is ${\left(\frac{2\beta }{3}\right)}^{\frac{3}{2}}\sqrt{T}.$
• C. Mean velocity of point averaged over the time taken by the point to move from the origin to the point where its x coordinate becomes $x_o$ is $\frac{2\beta x_o^{\frac{1}{3}}}{3}$
• D. All above statements are false.

Answer 1

Answer: (A), (B), (C)

Mean velocity of point averaged over the time taken by the point to move from the origin to the point where its x coordinate becomes $x_o$ is $\frac{2\beta x_o^{\frac{1}{3}}}{3}$

$V=\beta x^{\frac{1}{3}}=\frac{dx}{dt}$
$\Rightarrow \int \frac{dx}{x^{\frac{1}{3}}}=\int \beta dt$
$\Rightarrow \frac{3}{2}{x}^{\frac{2}{3}}=\beta t(\because x=0\text{at}t=0)$
$x={\left(\frac{2}{3}\beta t\right)}^{\frac{3}{2}}$
$V=\beta x^{\frac{1}{3}}$
Acceleration $a=\frac{dV}{dt}=V\frac{dV}{dx}$
$=\beta x^{\frac{1}{3}}\times \frac{1}{3}\beta x^{-\frac{2}{3}}=\frac{{\beta }^2}{3}{x}^{-1/3}$
∴ Acceleration is variable.
Average velocity upto time $T$,
$v_{avg}=\frac{\text{Total displacement}}{\text{Total time}}$
$=\frac{(\frac{2}{3}\beta T{)}^{\frac{3}{2}}}{T}={\left(\frac{2}{3}\beta \right)}^{\frac{3}{2}}\sqrt{T}$
$x={\left(\frac{2}{3}\beta t\right)}^{\frac{3}{2}}$
At $x=x_o\Rightarrow t=\frac{3x_o^{\frac{2}{3}}}{2\beta }$
Average velocity to $x_o$,
$v=\frac{\text{Total displacement}}{\text{Total time}}=\frac{x_o}{\frac{3x_o^{\frac{2}{3}}}{2\beta }}$
$=\frac{2\beta x_o^{\frac{1}{3}}}{3}$