Q 11-02-117JEE MainJEE Main 2020 (9 Jan, Shift 1)Medium
The distance $x$ covered by a particle in one dimensional motion varies with time $t$ as $x^2 = at^2 + 2bt + c$. If the acceleration of the particle depends on $x$ as $x^{-n}$, where $n$ is an integer, the value of $n$ is ______.
Numerical value type. Enter your answer.
Answer: 3
Differentiate: $2xv = 2at + 2b$, so $v = \dfrac{at + b}{x}$.
Differentiate again:
$$a_{\text{acc}} = \frac{a x - (at + b)v}{x^2} = \frac{ax^2 - (at+b)^2}{x^3}$$
The numerator is $a(at^2 + 2bt + c) - (at + b)^2 = ac - b^2$, a constant. So $a_{\text{acc}} \propto x^{-3}$ and $n = 3$.
Solution by Sreeraj P, M.Sc Physics