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Rec.Bicycles Frequently Asked Questions Posting Part 4/5
Section - 9.3 Road Gradient Units

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The grade of an incline is its vertical rise, in feet, per every 100 horizontal
feet traversed.  (I say "feet" for clarity;  one could use any consistent
length measure.)  Or, if you will accept my picture below,

                                                 *
                                              d  |
                                          a      |
                                      o          | y
                                  R  Theta       |
                              *___)______________|
                                        x
then
      Grade = y/x        (Multiply by 100 to express as a percentage.)
and
      Theta = arctan(y/x)

So a grade of 100% is a 45 degree angle.  A cliff has an infinite grade.

[More from Jobst Brandt <jobst.brandt@stanfordalumni.org>]
Date:    Mon, 26 Apr 1999 16:11:44 PDT

The steepness of a road is generally measured in % grade, which in
mathematical terms is the slope, or TANGENT of the angle, measured
from the horizontal.  This is the ratio of elevation change per
horizontal distance traveled, often called "rise over run".  Typically
a road that rises 1-in-10, is otherwise called 10% grade.

Measuring the distance along the surface of the road instead of
horizontally gives practically the same result for most road
gradients.  The distance along the road surface gives the SINE of the
angle in contrast to the horizontal distance that gives the TANGENT.
For practical purposes SINE equals TANGENT for small angles (up to ten
degrees or so).  For instance, a 20% grade (11.3 degrees), whereas
measuring along the road surface gives a 19.6% grade.

The slope of a road is more useful than its angle because it gives a
direct way to assess the effort required to move forward against the
grade, whereas the angle in degrees does not readily reveal this
information.  A 5% grade requires a forward force of approximately 5%
of the vehicle weight (above and beyond the force it takes to travel
similarly on flat ground).  A 15% grade requires a propulsion force of
approximately 15% of the vehicle weight.

Although the angle may be more easily visualized, it does not convert
easily to effort without a calculator.  For instance a 20% grade is an
11.3 degree angle and is a steep and difficult gradient.  The
relationship between angle and slope is non linear becoming 100% (1:1)
at a 45 degree angle.  In contrast, the SINE of 45 degrees is 70.7%
while the SINE of 90 degrees (straight up) is 100% for which the slope
(TANGENT) is infinity (or undefined).

The most accurate way to measure this without a precision
inclinometer, is to use a level, a one meter long bar and a metric
ruler.  Resting one end of the rod (held level) on the road at a
representative spot, measuring the distance down to the road at the
other end in centimeters gives the percent grade directly.  Using a
carpenters level and a one meter long rectangular bar can give
accurate readings to a couple of tenths of a percent.

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