Chapter 6.4, Problem 54E

(a)

To determine

To prove : The statement is $s(t)=\frac{v_{0}m}{k}\left(1-e^{-(k/m)t}\right)$

Explanation of Solution

Given information:

The velocity is $v=v_0{e}^{-(k/m)t}$

The statement is $s(t)=\frac{v_{0}m}{k}\left(1-e^{-(k/m)t}\right)$

Calculation:

The velocity equation is $v=v_0{e}^{-\frac{k}{m}t}$ .

  $\begin{array}{c}v=v_0{e}^{-\frac{k}{m}t}\\ \Rightarrow \frac{ds}{dt}=v_0{e}^{-\frac{k}{m}t}\\ \Rightarrow ds=v_0{e}^{-\frac{k}{m}t}dt\\ \Rightarrow \int ds=\int v_0{e}^{-\frac{k}{m}t}dt\\ \Rightarrow s=v_0\int e^{-\frac{k}{m}t}dt\\ \Rightarrow s=v_0\left(\frac{e^{-\frac{k}{m}t}}{-\frac{k}{m}}\right)+C\\ =v_0\left(-\frac{m}{k}\right)e^{-\frac{k}{m}t}+C\\ =-\frac{v_{0}m}{k}{e}^{-\frac{k}{m}t}+C\end{array}$

Given that,

  $s(0)=0$

So, put in

  $\begin{array}{c}s=0\\ \text{t=0}\end{array}$

To find $\text{C}$ :

  $\begin{array}{c}s=-\frac{v_{0}m}{k}{e}^{-\frac{k}{m}t}+C\\ \Rightarrow 0=-\frac{v_{0}m}{k}{e}^{-\frac{k}{m}(0)}+C\\ \Rightarrow 0=-\frac{v_{0}m}{k}{e}^0+C\\ \Rightarrow 0=-\frac{v_{0}m}{k}+C\\ \Rightarrow \frac{v_{0}m}{k}=C\\ \Rightarrow s=-\frac{v_{0}m}{k}{e}^{-\frac{k}{m}t}+\frac{v_{0}m}{k}\left(-e^{-\frac{k}{m}t}+1\right)\\ =\frac{v_{0}m}{k}{\left(1-e^{-\frac{k}{m}t}\right)}^n\end{array}$

Therefore, the required given statement $s(t)=\frac{v_{0}m}{k}\left(1-e^{-(k/m)t}\right)$ is $($ proved $)$ .

(b)

To determine

To prove: The given statement.

Explanation of Solution

Given information:

The velocity is $v=v_0{e}^{-(k/m)t}$

The total coasting distance traveled by the object as its coasts to the complete stop is $v_{0}m/k$

Calculation:

The velocity of the object is $v=v_0{e}^{-\frac{k}{m}t}$ which never equals 0 but gets infinitesimally close to 0 since the function has a horizontal asymptote at $v=0$ . The total coasting distance is then the limit of the distance function as $t\to \infty$ Find the limit using $e^{-\infty }\to 0$ .

  $\begin{array}{c}\underset{t\to \infty }{\lim }s(t)=\underset{t\to \infty }{\lim }\frac{v_{0}m}{k}\left(1-e^{-\frac{k}{m}t}\right)\\ =\frac{v_{0}m}{k}\left(1-e^{-\infty }\right)\\ =\frac{v_{0}m}{k}(1-0)\\ =\frac{v_{0}m}{k}\end{array}$

Therefore, the required given statement is $($ proved $)$ .