Question

A particle located at x=0, starts moving along the position x-direction with a velocity v that as $ v = a\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

Hint: Displacement is the vector difference between the ending and starting positions of an object.
It may be very different from the distance the object has travelled along the way.
The unit of velocity is $ m/s $ whereas the $ SI $ unit of displacement is meter.

Complete step by step answer:
Velocity, $ v = a\sqrt x \left( {\because v = \dfrac{{dx}}{{dt}}} \right) $
 $ \dfrac{{dx}}{{\sqrt x }} = adt $
Perform integration
 $ \smallint _0^x\dfrac{{dx}}{{\sqrt x }} = \smallint _0^tadt $
 $ \left( {\because t = 0,x = 0} \right) $ and let at any time $ t $ , particle is at $ x $
 $
   \Rightarrow \dfrac{{{x^{1/2}}}}{{1/2}}|_0^x = at \\
   \Rightarrow {x^{1/2}} = \dfrac{a}{2}t \\
   \Rightarrow x = \dfrac{{{a^2}}}{4} \times {t^2} \\
   \therefore x\alpha {t^2} \\
 $

So, the correct answer is “Option A”.

Note:
Displacement is the vector difference between the ending and starting positions of an object. It may be very different from an object that has travelled along the way. Velocity is the rate at which displacement changes with time. It is a vector, too. The average velocity over some interval, divided by the time. The only difference between velocity and speed is that velocity has a direction, and so is a vector quantity. Displacement is the distance moved in a straight line, in a given direction, from the starting point. Displacement is a vector quantity as it has size and direction. Displacement is the difference in distance between the starting and final position. So speed depends on distance travelled, while velocity depends on the displacement.
Velocity $ = $ displacement (vector from initial position to final position) $ / $ time elapsed in making this displacement. Displacement is total path length. Displacement and hence velocity can be positive or negative depending on the chosen coordinate system, but distance and hence speed are always positive.