Question

The displacement of the particle is found to be varying with respect to the time according to the relation $x=\dfrac{k}{b}\left[ 1-{{e}^{-bt}} \right]$. Then what will be the velocity of the particle?
$\begin{align}
  & A.V=k{{e}^{\left( -bt \right)}} \\
 & B.V=\dfrac{k}{b}{{e}^{\left( -bt \right)}} \\
 & C.V=\dfrac{{{k}^{2}}}{b}{{e}^{\left( -bt \right)}} \\
 & D.V=\dfrac{k}{{{b}^{2}}}{{e}^{\left( -bt \right)}} \\
\end{align}$

Answer 1

Hint: The acceleration of a body can be found by taking the rate of the variation of the velocity with respect to the time taken. The velocity can be found by taking the rate of the variation of the displacement with respect to the time taken. The displacement of the body is the perpendicular distance between the initial and final locations of the body. All these quantities are vector quantities possessing both magnitudes as well as direction.

Complete step-by-step solution
the acceleration of a body can be found by taking the rate of the variation of the velocity with respect to the time taken. The velocity can be found by taking the rate of the variation of the displacement with respect to the time taken. The displacement of the body is the perpendicular distance between the initial and final locations of the body. All these quantities are vector quantities possessing both magnitudes as well as direction.
It has been mentioned in the question that the displacement will be,
$x=\dfrac{k}{b}\left[ 1-{{e}^{-bt}} \right]$
Therefore the velocity of the wave will be the first derivative of the displacement. That is,
$\begin{align}
  & V=\dfrac{dx}{dt}=\dfrac{d}{dt}\left( \dfrac{k}{b}\left( 1-{{e}^{-bt}} \right) \right) \\
 & V=\dfrac{k}{b}\left( 0-\left( -b \right){{e}^{-bt}} \right) \\
 & \therefore V=k{{e}^{-bt}} \\
\end{align}$
Therefore the velocity of the wave has been calculated. Hence the correct answer is option A.

Note: There are two kinds of acceleration in general. They are average acceleration and instantaneous acceleration. An average acceleration of the object over a period of time will be its variation in velocity divided by the total duration of the period. Instantaneous acceleration will be the limit of the average acceleration over an infinitesimal interval of time. The instantaneous acceleration will be the derivative of the velocity vector with respect to time in terms of calculus.