Read Online X and the City: Modeling Aspects of Urban Life by John A. Adam - Free Book Online
Authors: John A. Adam
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It is readily confirmed that this has a maximum value of about 0.76 liters when
θ
≈ 82.4°. A graph of
T
(
θ
) is shown in Figure 6.3 for −
π
/2 <
θ
<
π
/2. Note also that for a given speed when
θ
< 0 (i.e., the rain is coming from behind you) the quantity of rain accumulated is smaller (as would be expected) than when you are running into the rain. Notice also the wide range of values, from a high of just below 0.8 liters to a low of less than 0.1 liters when the rain is hitting you horizontally from behind!
We need to be a little more careful with the case of
θ
< 0 because (6.2) could become negative (and therefore meaningless) for some parameter ranges. If we replace
θ
by −,> 0, then this equation becomes
This is negative if
Figure 6.3. Total amount of rain captured as a function of rain angle (radians).
Note that this can never happen if
v
≥
c
, and will not for the choice of
v
and
c
made here (
v
/
c
= 1.2). If the inequality (6.4)
is
satisfied, the “offending term” comes from the “front accumulation” value
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, andin 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!
X
= Δ
T
: WEATHER IN THE CITY
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 countrysideis further enhanced at night, because the latter loses more heat by evaporative and other processes. Furthermore, the combined effects of
θ
≈ 82.4°. A graph of
T
(
θ
) is shown in Figure 6.3 for −
π
/2 <
θ
<
π
/2. Note also that for a given speed when
θ
< 0 (i.e., the rain is coming from behind you) the quantity of rain accumulated is smaller (as would be expected) than when you are running into the rain. Notice also the wide range of values, from a high of just below 0.8 liters to a low of less than 0.1 liters when the rain is hitting you horizontally from behind!
We need to be a little more careful with the case of
θ
< 0 because (6.2) could become negative (and therefore meaningless) for some parameter ranges. If we replace
θ
by −,> 0, then this equation becomes
This is negative if
Figure 6.3. Total amount of rain captured as a function of rain angle (radians).
Note that this can never happen if
v
≥
c
, and will not for the choice of
v
and
c
made here (
v
/
c
= 1.2). If the inequality (6.4)
is
satisfied, the “offending term” comes from the “front accumulation” value
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, andin 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!
X
= Δ
T
: WEATHER IN THE CITY
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 countrysideis further enhanced at night, because the latter loses more heat by evaporative and other processes. Furthermore, the combined effects of
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