X and the City: Modeling Aspects of Urban Life (17 page)

which should now be written as

because the rain falls on your back if sin
v
<
c
sin
. The correct total is now

It is instructive to rewrite this equation as

Let’s focus our attention on the term in parentheses in equation (6.6), noting that
t
/
l
=
A
_top
/
A
_back
is just the ratio of the top area of the human “box” to the area of the back (or front). If tan
>
t
/
l
=
A
_top
/
A
_back
, this term is negative, and
in this case, you should attempt to go no faster than the horizontal speed of the rain (
c
sin
) at your back. Using equation (6.5) we see that if your speed increases so that
v
=
c
sin
, you are just keeping up with the rain and
T
is minimized. This may seem at first surprising since for
v
>
c
sin
,
T
is reduced still farther, but
now
you are catching up to the rain ahead of you, and it falls once more on your front (and head, of course). In this case formula (6.2) again applies.

How about putting some numbers into these formulae? For a generic height
l
= 175 cm (about 5 ft 9 in), shoulder to shoulder width
w
= 45 cm (about 18 in), and chest to back width
t
= 25 cm (about 10 in), the ratio
t
/
l
= 1/7, and so if tan
> 1/7, that is,
≈ 8°, the ratio
v
/
c
= sin
≈ 1/7 (see why?). Therefore if it is raining heavily at about 5 m/s from this small angle to the vertical, you need only amble at less than one m/s (about 2 mph) to minimize your accumulated wetness! Although the chosen value for
w
was not used, we shall do so now. The top area of our human box is ≈ 1100 cm
²
, one side area is ≈ 4400 cm
²
, and the front or back area is ≈ 7900 cm
²
.

Exercise:
Calculate these areas in square feet if you feel so inclined.

To summarize our results, if the rain is driving into you from the front, run as fast as you safely can. On the other hand, if the rain is coming from behind you, and you can keep pace with its horizontal speed by walking, do so! If you exceed that speed, the advantage of getting to your destination more quickly is outweighed by the extra rain that hits you from the front, since you are now running into it! Perhaps the moral of this is that we should always run such that the rain is coming from behind us!

To some extent cities can create their own weather. No doubt you have heard of the sidewalk in some city being hot enough to fry an egg; include all the paved surfaces and buildings in a city, and you have the capacity to cook a lot of breakfasts! Typically, such surfaces get hotter than those in rural environments because they absorb more solar heat (and therefore reflect less), and retain that heat for longer than their rural counterparts by virtue of their greater thermal “capacity.” The contrast between a city and the surrounding countryside
is further enhanced at night, because the latter loses more heat by evaporative and other processes. Furthermore, the combined effects of traffic and industrial plants are a considerable source of heat within an urban metropolitan area. Thus there are several factors to take into account when considering local climate in a city versus that in the countryside. They include the fact that (i) there are differences between surface materials in the city and the countryside—concrete, tarmac, soil, and vegetation; (ii) the city “landscape”—roofs, walls, sidewalks, and roads—is much more varied than that in the country in the shape and orientation of reflective surfaces; vertical walls tend to reflect solar radiation downward instead of skyward (see
Figure 6.4a,b
), and concrete retains heat longer than do soil and vegetation; (iii) cities are superb generators of heat, particularly in the winter months; (iv) cities dispose of precipitation in very different ways, via drains, sewers, and snowplows (in the north). In the country, water and snow are more readily available for evaporative cooling.