As I was driving on the highway this afternoon, I thought to myself: what if, at each moment, I were to move at a speed that matched exactly the distance I had remaining? As an example, at 60 miles from the destination I would drive at 60 miles per hour; then, with 59 miles remaining, I would slow to 59 miles per hour; and so on, of course with infinitesimal precision. Two questions emerge from this situation:
- How long will it take to travel from mile-marker 60 to mile-marker 0?
- And how long will it take to travel from mile-marker 60 to, say, mile-marker 20?
I've read through several previous postings of this same question (here and here, for example) but have yet to find a satisfying, thorough explanation. (How exactly, for example, do we involve the harmonic number $H_{60}$ to this problem?)
Your velocity is $v(t)=60-x(t)$ where $x(t)$ is the distance from start. Note that $v(t)=x'(t)$ so this is $x'(t)=60-x(t)$. Solution to this differential equation is $x(t)=60+Ce^{-t}$. Note that $x(0)=0$ and so $x(t)=60-60e^{-t}$. Note that for all finite time $t$ we have $x(t)<60$ (i.e. you aren't at your desination). However $\lim_{t\to\infty} x(t)=60$.