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Up: The Exponential Function Previous: Summary: The Exponential Function(s)

An Example from Mechanics: Damping

We should really work out at least one example applying the exponential function to a real Mechanics problem. The classic example is where an object (mass  m) is moving with an initial velocity  v0,  starting from an initial position  x0,  and experiences a frictional damping force  Fd  which is proportional to the velocity and (as always, for frictional forces) in the direction opposite to the velocity:   $F_d = - \kappa \, v$. The equation of motion then reads   $a = - (\kappa/m) \, v$  or

\begin{displaymath}{d^2 x \over dt^2} \; = \; - k \, {dx \over dt}
\end{displaymath} (27)

where we have combined  $\kappa$  and  m  into the constant   $k \equiv \kappa/m$. This can also be written in the form

\begin{displaymath}{dv \over dt} \; = \; - k \, v
\end{displaymath}

which should ring a bell! The solution (for the velocity  v) is

\begin{displaymath}v(t) \; = \; v_0 \, e^{- k \, t}
\end{displaymath} (28)

To obtain the solution for  x(t),  we switch back to the notation

\begin{displaymath}{dx \over dt} = v_0 \, e^{- k \, t}
\qquad \Longrightarrow \qquad
\int_{x_0}^x dx = v_0 \int_0^t e^{- k \, t} \; dt
\end{displaymath}

and note that the function whose time derivative is   $e^{-k \, t}$  is   $-{1 \over k} \, e^{-k \, t}$,  giving

\begin{displaymath}x - x_0 \; = \; - {v_0 \over k} \left[ e^{-k \, t} \right]_0^t
\end{displaymath}

where the   $[\cdots]_0^t$  notation means that the expression in the square brackets is to be ``evaluated between 0 and t'' -- i.e. plug in the upper limit (just t itself) for t in the expression and then subtract the value of the expression with the lower limit (0) substituted for t. In this case the lower limit gives   e-0 = e0 = 1  (anything to the zeroth power gives one) so the result is

\begin{displaymath}x(t) \; = \; x_0 \; + \; {v_0 \over k} \,
\left( 1 - e^{- k \, t} \right)
\end{displaymath} (29)

The qualitative behaviour is plotted in Fig. 2. Note that  x(t)  approaches a fixed ``asymptotic'' value   $x_{\rm max} = x_0 + v_0/k$  as   $t \to \infty$. The generic function   $(1 - e^{-k \, t})$  is another useful addition to your pattern-recognition repertoire.


  
Figure: Solution to the damping force equation of motion, in which the frictional force is proportional to the velocity.
\begin{figure}
\begin{center}\mbox{
\epsfig{file=PS/damp.ps,height=5.0in} }\end{center}\end{figure}


next up previous
Up: The Exponential Function Previous: Summary: The Exponential Function(s)
Jess H. Brewer
1998-08-04