Question

A particle located at $ x = 0 $ at time $ t = 0 $ , starts moving along the positive $ x $ -direction with a velocity $ v $ that varies as $ v = \alpha \sqrt {x} $ . The displacement of the particle varies with time as

• A. $ t^2 $
• B. $ t $
• C. $ t^{1/2} $
• D. $ t^3 $

Answer 1

Answer: (A)

Velocity, $v=\alpha \sqrt{x}$
$\frac{d x}{d t}=\alpha \sqrt{x} \left(\because v=\frac{d x}{d t}\right)$
$\frac{d x}{\sqrt{x}}=\alpha d t$
Perform integration
$\int_{0}^{x} \frac{d x}{\sqrt{x}}=\int_{0}^{t} \alpha d t$
$(\because$ at $t=0, x=0$ and let at any time $t$, particle is at $x$ )
$\Rightarrow \left.\frac{x^{1 / 2}}{1 / 2}\right|_{0} ^{x}=\alpha t$
$\Rightarrow x^{1 / 2}=\frac{\alpha}{2} t$
$\Rightarrow x=\frac{\alpha^{2}}{4} \times t^{2}$
$\Rightarrow x \propto t^{2}$