Question

The velocity of a particle moving in the positive direction of $x-$ axis varies as $v=10\sqrt{x}$ ($v$ is in ${\text{ms}}^{-1},x$ is in $m$). Assuming that at $t=0,$ particle was at $x=0$. Then,

• A. The initial velocity of the particle is zero
• B. The initial velocity of the particle is $2.5{\text{ms}}^{-1}$
• C. The acceleration of the particle is $2.5{\text{ms}}^{-2}$
• D. The acceleration of the particle is $50{\text{ms}}^{-2}$

Answer 1

Answer: (A), (D)

The correct options are
A The initial velocity of the particle is zero
D The acceleration of the particle is $50{\text{ms}}^{-2}$

Given $v=10\sqrt{x}.......\left(1\right)$
$\frac{dx}{dt}=10\sqrt{x}\Rightarrow \int \frac{dx}{\sqrt{x}}=10\int dt$
$\Rightarrow 2\sqrt{x}=10t+c$
Given that at $t=0,x=0$,
$\therefore c=0$
$\therefore 2\sqrt{x}=10t\Rightarrow x=25t^2$
Substituting in $\left(1\right)$,
$v=50t$
$\therefore \text{At}t=0,v=0$
Differentiating $v$
$a=\frac{dv}{dt}=50{\text{ms}}^{-2}$