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. $x^2=a{t}^2+2bt+c$

differentiating w. r. t. time,

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

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

Again, differentiating w. r. t. time,

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

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

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

$\Rightarrow {\text{acc}}^n\propto x^{-3}$

Hence, $n=3$