Question

A particle is moving with speed $v=b\sqrt{x}$ along the positive $x$ - axis. Calculate the speed of the particle at time $t=\tau$. (Assume that the particle is at the origin at $t=0$).

• A. $\frac{b^2\tau }{4}$
• B. $\frac{b^2\tau }{2}$
• C. $b^2\tau$
• D. $\frac{b^2\tau }{\sqrt{2}}$

Answer 1

Answer: (B)

$\frac{b^2\tau }{2}$

Given, $\begin{array}{}v=b\sqrt{x}& \text{(1)}\end{array}$
$\frac{dx}{dt}=b\sqrt{x}$
${\int }_0^x\frac{dx}{\sqrt{x}}={\int }_0^{t}bdt$
$\Rightarrow 2\sqrt{x}=bt$
$\Rightarrow \sqrt{x}=\frac{bt}{2}$
Substituting $\sqrt{x}$ in equation $\left(1\right)$, we get
$\Rightarrow v=\frac{b^{2}t}{2}$
At $t=\tau ,v=\frac{b^2\tau }{2}$