The distance problem

If I want to know how far I walked during an hour, I can ask how far I walked in the first five minutes, and how far I walked in the second five minutes, and how far I walked in the third five minutes, etc. and add them all together. ie. I could write:

$d=d_1+d_2+d_3+d_4+...d_{12}$

Where $d_i$ is the distance walked in the $i^{th}$ five minutes. To calculate a distance, we need to know how fast we are going, and for how long. In fact:

$distance=velocity \times time$

where you can think of velocity as the same thing as speed (though there are subtle differences which you will find out about later). This formula works if the velocity is constant, but what if it is changing. Well, if we have a graph of velocity against time, then we can think about splitting the graph into intervals (like the five minute intervals above), and approximating that during a small interval of time, the velocity is roughly constant. That is the same thing as approximating a curve by a series of rectangles that we did in the last post. Now we see that if we approximate like this, the distance traveled is simply the area under the rectangles, which, as we take smaller and smaller rectangles, better and better approximates the area under the curve.

If we write the velocity as a function of time as $v(t)$, then we can write:

$d\approx \sum_i v(t_i)\Delta t$

and the distance is best approximated when we take the limit:

$d = \lim_{n\rightarrow \infty} \sum_i v(t_i)\Delta t$

Where $\Delta t=\frac{t_f-t_i}{n}$, where $t_i$ is the start time, and $t_f$ is the end time, and we have split it into n intervals.

We can see how the distance problem and the area problem are really part of the same thing, and the same techniques will be used for both of them.

 How clear is this post?