Question

The distance $x$ covered by a particle in one dimensional motion varies with time $t$ as $x^2=a{t}^2+2bt+c$. If the acceleration of the particle depends on $x$ as $x^{-n}$, where $n$ is an integer, the value of $n$ is ____.

Answer 1

Distance $x$ varies with time $t$ as,

$x^2=a{t}^2+2bt+c...\left(1\right)$

Differentiating both side with respect to t,

$\Rightarrow 2x\frac{dx}{dt}=2at+2b$

$\Rightarrow x\frac{dx}{dt}=at+b...\left(2\right)$

Again differentiating,

$\Rightarrow x\frac{d^{2}x}{d{t}^2}+{\left(\frac{dx}{dt}\right)}^2=a$

$\Rightarrow \frac{d^{2}x}{d{t}^2}=\frac{a-{\left(\frac{dx}{dt}\right)}^2}{x}$

Using equation $\left(2\right)$,

$\Rightarrow \frac{d^{2}x}{d{t}^2}=\frac{a-{\left(\frac{at+b}{x}\right)}^2}{x}$

$\Rightarrow a=\frac{d^{2}x}{d{t}^2}=\frac{a{x}^2-(at+b{)}^2}{x^3}$

Using equation $\left(1\right)$,

$\Rightarrow a=\frac{ac-b^2}{x^3}$

$\Rightarrow a\propto x^{-3}$

$\text{Hence, n=3}$