Question

The motion of a particle is described by \[x = {x_0}\left( {1 - {e^{ - kt}}} \right)\]; \[t \geqslant 0\]; \[{x_0} > 0\]; \[k > 0\]. With what velocity does the particle start?
A. \[\dfrac{{{x_0}}}{k}\]
B. \[{x_0}k\]
C. \[\dfrac{k}{{{x_0}}}\]
D. \[2{x_0}k\]

Answer 1

Hint: Use the formula for velocity of a particle in terms of its displacement and time required for the displacement. Substitute the value of the given equation of displacement in this formula and derive the relation for velocity of the particle. Substitute 0 for time in this equation of velocity and determine the value of the initial velocity of the particle.

Formula used:
The velocity \[v\] of a particle is given by
\[v = \dfrac{{dx}}{{dt}}\] …… (1)
Here, \[dx\] is a change in displacement of the particle in time interval \[dt\].

Complete step by step answer:
We have given that the equation of motion of a particle is given by
\[x = {x_0}\left( {1 - {e^{ - kt}}} \right)\]
In the above equation, \[t \geqslant 0\], \[{x_0} > 0\] and \[k > 0\].
We have asked to determine the velocity with which the particle starts its motion that initial velocity of the particle. Let us first determine the equation for velocity of the particle using equation (1).
Substitute \[{x_0}\left( {1 - {e^{ - kt}}} \right)\] for \[x\] in equation (1).
\[v = \dfrac{{d\left[ {{x_0}\left( {1 - {e^{ - kt}}} \right)} \right]}}{{dt}}\]
\[ \Rightarrow v = {x_0}\left( { - {e^{ - kt}}} \right)\left( { - k} \right)\]
\[ \Rightarrow v = k{x_0}{e^{ - kt}}\]
This is the expression for velocity of the particle in motion.

Let \[{v_0}\] be the initial velocity of the particle.
When the particle starts its motion, the time \[t\] is \[t = 0\,{\text{s}}\].
Substitute \[{v_0}\] for \[v\] and \[0\,{\text{s}}\] for \[t\] in the above equation of velocity.
\[ \Rightarrow {v_0} = {x_0}k{e^{ - k\left( {0\,{\text{s}}} \right)}}\]
\[ \Rightarrow {v_0} = {x_0}k{e^0}\]
\[ \Rightarrow {v_0} = {x_0}k\left( 1 \right)\]
\[ \therefore {v_0} = {x_0}k\]
Therefore, the velocity with which the particle starts its motion is \[{x_0}k\].

Hence, the correct option is B.

Note: The students should be careful while taking the integration of displacement with respect to time to derive the formula for velocity of the particle. If this integration is not taken properly, the derivation of velocity of the particle will be incorrect and hence the value of initial velocity found will also be incorrect.