Question

A particle moves rectilinearly. Its displacement $x$ at time $t$ is given by $x^2=a{t}^2+b$ where $a$ and $b$ are constants. Its acceleration at time $t$ is proportional to

• A. $\frac{1}{x^3}$
• B. $\frac{1}{x}-\frac{1}{x^2}$
• C. $-\frac{t}{x^2}$
• D. $\frac{1}{x}-\frac{t^2}{x^3}$

Answer 1

Answer: (A)

Given : $x^2=a{t}^2+b\dots \left(i\right)$
Differentiating w.r.t. $t$ on both sides, we get
$2x\frac{dx}{dt}=2at$ ; $xv=at\left(\because v=\frac{dx}{dt}\right)$
Again differentiating w.r.t. $t$ on both sides, we get
$x\frac{dv}{dt}+v\frac{dx}{dt}=a$ or $x\frac{dv}{dt}=a-v^2$
$\frac{dv}{dt}=\frac{a-v^2}{x}=\frac{a-{\left(\frac{at}{x}\right)}^2}{x}$
$=\frac{a-\frac{a^2{t}^2}{x^2}}{x}=\frac{a\left(x^2-a{t}^2\right)}{x^3}$
$\frac{dv}{dt}=\frac{ab}{x^3}$ or Acceleration $\propto \frac{1}{x^3}$ (Using $(i)$)