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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  F_d  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

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

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

$${dv \over dt} \; = \; - k \, v$$

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

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

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

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

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

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

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

$$x(t) \; = \; x_0 \; + \; {v_0 \over k} \, \left( 1 - e^{- k \, t} \right)$$ (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.