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. ${t^3}$
B. ${t^2}$
C. $t$
D. ${t^{\dfrac{1}{2}}}$

Answer 1

We are asked to find how the displacement of the particle is dependent on time. We are given the velocity of the particle as a function of its displacement, so try to write velocity in terms of time and displacement, then use the equation to find the relation between displacement and time.

Complete step by step answer:
Given, initial position $x = 0$ at time $t = 0$. Velocity of the particle, $v = \alpha \sqrt x$. Let the displacement of the particle be $x$. Velocity can be defined as the rate of change of position. That is we can write velocity as,
$v = \dfrac{{dx}}{{dt}}$
where $dx$ is the change in position in time interval $dt$.
Putting the value of $v$, we get
$\alpha \sqrt x = \dfrac{{dx}}{{dt}}$
$\Rightarrow \dfrac{{dx}}{{\sqrt x }} = \alpha dt$
Now, integrating from $x = 0$ to $x$ on left hand side and $t = 0$ to $t$ on right hand side, we get
$\int\limits_{x = 0}^x {\dfrac{{dx}}{{\sqrt x }}} = \int\limits_{t = 0}^t {\alpha dt}$
$\Rightarrow \int\limits_{x = 0}^x {{x^{ - \dfrac{1}{2}}}dx} = \alpha \int\limits_{t = 0}^t {dt}$
$\Rightarrow \left[ {\dfrac{{{x^{ - \dfrac{1}{2} + 1}}}}{{^{ - \dfrac{1}{2} + 1}}}} \right]_{x = 0}^x = \alpha \left[ t \right]_{t = 0}^t$
$\Rightarrow \dfrac{{{x^{\dfrac{1}{2}}}}}{{\left( {\dfrac{1}{2}} \right)}} = \alpha t$
$\Rightarrow {x^{\dfrac{1}{2}}} = \dfrac{1}{2}\alpha t$
Squaring both sides we get,
${\left( {{x^{\dfrac{1}{2}}}} \right)^2} = {\left( {\dfrac{1}{2}\alpha t} \right)^2}$
$\Rightarrow x = \dfrac{{{\alpha ^2}{t^2}}}{4}$
From the above equation, we observe that
$\therefore x \propto {t^2}$
Therefore, the displacement of the particle varies as ${t^2}$.

Hence, the correct answer is option B.

Note: We have discussed velocity . Velocity is the rate of change of position of a body. If there is no change in position of a body in a given interval then its velocity is zero. Similarly, the rate of change of velocity is known as acceleration and the body is moving with constant velocity then its acceleration is zero or we can say the body is moving with uniform velocity.