Question

A particle moves along a straight line and at a distance $x$ from a fixed origin $O$ on the line. Its velocity is proportional to $\sqrt{\frac{\beta -\alpha }{x}}$. Then, its acceleration is

• A. directed towards $O$ and is proportional to $1/x$.
• B. directed towards $O$ and is proportional to $1/x^2$.
• C. directed towards $O$ and is proportional to $x$.
• D. directed away from $O$ and is proportional to $x^2$.

Answer 1

The correct option is B directed towards $O$ and is proportional to $1/x^2$.

$v\propto \sqrt{\frac{\beta -\alpha }{x}}\Rightarrow v\alpha \frac{1}{\sqrt{x}}a=v\frac{dv}{dx}\Rightarrow a\propto -\frac{1}{\sqrt{x}}\times \frac{1}{x\sqrt{x}}=-\frac{1}{x^2}$

as acceleration is opposite direction of position, it is directed towards origin.