Answer

Verified

444.9k+ views

**Hint:**We are given the expression for velocity of the particle as a function of time. The average velocity of a particle is equal to the integral of velocity with respect to time divided by the integral of time. By integrating up to the given duration of time we can obtain the required answer.

**Formula used:**

The average velocity of a particle can be calculated by the following formula:

${v_{av}} = \dfrac{{\int\limits_{{t_1}}^{{t_2}} {vdt} }}{{\int\limits_{{t_1}}^{{t_2}} {dt} }}$

**Complete step-by-step solution**Detailed step by step solution:

We are given a particle whose velocity is given as a function of time by the following expression:

\[v = {v_0}{e^{ - bt}}\]

Here ${v_0}$ and b are constants.

We need to calculate the average velocity of the particle from t = 0 up to $t = \dfrac{1}{b}$. This can be done by using the formula for the average velocity of the particle in the following way.

${v_{av}} = \dfrac{{\int\limits_0^t {vdt} }}{{\int\limits_0^t {dt} }}$

Now we will insert the value of velocity in the above expression. Doing so, we get

${v_{av}} = \dfrac{{\int\limits_0^t {{v_0}{e^{ - bt}}dt} }}{{\int\limits_0^t {dt} }}$

Now we will integrate the numerator and the denominator. Doing so. we get the following expression.

$\begin{align}

&{v_{av}} = \dfrac{{{v_0}\int\limits_0^t {{e^{ - bt}}dt} }}{{\int\limits_0^t {dt} }} = \dfrac{{{v_0}\left[ {\dfrac{{{e^{ - bt}}}}{{ - b}}} \right]_0^t}}{{\left[ t \right]_0^t}} \\

&= - \dfrac{{{v_0}}}{b}\dfrac{{\left( {{e^{ - bt}} - 1} \right)}}{t} \\

\end{align} $

Now we will insert the given value of t which is $t = \dfrac{1}{b}$. Doing so, we obtain our required answer for the average velocity of the particle.

$\begin{align}

&{v_{av}} = - \dfrac{{{v_0}}}{b}\dfrac{{\left( {{e^{ - b \times \dfrac{1}{b}}} - 1} \right)}}{{\dfrac{1}{b}}} \\

& = - {v_0}\left( {{e^{ - 1}} - 1} \right) \\

& = - {v_0}\left( {\dfrac{1}{e} - 1} \right) \\

& = - {v_0}\left( {\dfrac{{1 - e}}{e}} \right) \\

& = {v_0}\left( {\dfrac{{e - 1}}{e}} \right) \\

\end{align} $

**Therefore, we can say that the correct answer is option C.**

**Note:**1. This formula for average velocity is derived from the elementary formula for average velocity according to which the average velocity is equal to the total distance traveled divided by the total time taken. The numerator in the formula represents the total distance which is divided by the total time.

2. The average velocity of a particle does not give complete information about the motion of the particle due to which we can also calculate the instantaneous velocity by directly inserting the value of time at which we need the velocity of the particle in the expression for velocity.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

The states of India which do not have an International class 10 social science CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

How do you graph the function fx 4x class 9 maths CBSE

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Why is there a time difference of about 5 hours between class 10 social science CBSE

Name the three parallel ranges of the Himalayas Describe class 9 social science CBSE