Question

The velocity of a particle is given by $$v=v_0{e}^{-bt}$$, where $v_0$ and b are constants. The average velocity of particle for the time duration $t={\displaystyle \frac{1}{b}}$ from start of its motion
$\begin{array}{rl}& \text{A}\text{. }{v}_{0}b\left[{\displaystyle \frac{1+e}{e}}\right]\\ & \text{B}\text{. }{\displaystyle \frac{v_0}{b}}\left[{\displaystyle \frac{e-1}{e}}\right]\\ & \text{C}\text{. }{v}_0\left[{\displaystyle \frac{e-1}{e}}\right]\\ & \text{D}\text{. }{v}_0\left[{\displaystyle \frac{e+1}{e}}\right]\end{array}$

Answer 1

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:
$v_{av}={\displaystyle \frac{\underset{t_1}{\overset{t_2}{\int }}vdt}{\underset{t_1}{\overset{t_2}{\int }}dt}}$

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 $t={\displaystyle \frac{1}{b}}$. This can be done by using the formula for the average velocity of the particle in the following way.
$v_{av}={\displaystyle \frac{\underset{0}{\overset{t}{\int }}vdt}{\underset{0}{\overset{t}{\int }}dt}}$
Now we will insert the value of velocity in the above expression. Doing so, we get
$v_{av}={\displaystyle \frac{\underset{0}{\overset{t}{\int }}{v}_0{e}^{-bt}dt}{\underset{0}{\overset{t}{\int }}dt}}$
Now we will integrate the numerator and the denominator. Doing so. we get the following expression.
$\begin{array}{rl}& v_{av}={\displaystyle \frac{v_0\underset{0}{\overset{t}{\int }}{e}^{-bt}dt}{\underset{0}{\overset{t}{\int }}dt}}={\displaystyle \frac{v_0{\left[{\displaystyle \frac{e^{-bt}}{-b}}\right]}_0^t}{{\left[t\right]}_0^t}}\\ & =-{\displaystyle \frac{v_0}{b}}{\displaystyle \frac{\left(e^{-bt}-1\right)}{t}}\end{array}$
Now we will insert the given value of t which is $t={\displaystyle \frac{1}{b}}$. Doing so, we obtain our required answer for the average velocity of the particle.
$\begin{array}{rl}& v_{av}=-{\displaystyle \frac{v_0}{b}}{\displaystyle \frac{\left(e^{-b\times {\displaystyle \frac{1}{b}}}-1\right)}{{\displaystyle \frac{1}{b}}}}\\ & =-v_0\left(e^{-1}-1\right)\\ & =-v_0\left({\displaystyle \frac{1}{e}}-1\right)\\ & =-v_0\left({\displaystyle \frac{1-e}{e}}\right)\\ & =v_0\left({\displaystyle \frac{e-1}{e}}\right)\end{array}$
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.