Vertical Climbing Speed

Hi all. I would love some of the experienced math and physics gurus
here to review and critique my analysis of vertical climbing rate
(with application to cycling). If you have any insights or how I could
improve, that'd be great. http://tinyurl.com/nu2p5w

Yeah. I did beat it to death. Now to find out if I beat the right one
to death.



BD

On Aug 1, 2:13*pm, bicycle_disciple wrote:
Hi all. I would love some of the experienced math and physics gurus
here to review and critique my analysis of vertical climbing rate
(with application to cycling). If you have any insights or how I could
improve, that'd be great.http://tinyurl.com/nu2p5w

Yeah. I did beat it to death. Now to find out if I beat the right one
to death.

BD

I don't know about all the maths. I just go by the rule of thumb that
a watt of output will raise one kg a thousand feet in an hour. So, if
Tom Danielson and his bike weighed 100kg total, he would have had to
put out a minimum of 540 watts continuous.

On 2009-08-01, bicycle_disciple wrote:
Hi all. I would love some of the experienced math and physics gurus
here to review and critique my analysis of vertical climbing rate
(with application to cycling). If you have any insights or how I could
improve, that'd be great. http://tinyurl.com/nu2p5w

Yeah. I did beat it to death. Now to find out if I beat the right one
to death.

When you say Rise/Run, by "Run" you mean distance travelled along the
road as measured by your cycle's odometer for example? If so it looks
right.

In the section "Practical and Theoretical Limits of Climbing Rate", we
have a graph of climbing rate against grade for a road-speed of 1. So if
my road-speed is 1, and the grade is very steep-- practically vertical--
then my VAM is almost 1, which is why we approach that 1 line at the top
and bottom. Vertical speed equals road speed for a vertical road, and
can never be greater than road speed of course.

So far so good, but Note 2 is a bit strange. Grade can't keep increasing
without bounds-- it can never get steeper than 1. For a constant road
speed, then the steeper the grade the higher the VAM which may be what
Ferrari is saying.

Whether you roll backwards or fall or your gears aren't low enough
doesn't really have anything to do with efficiency. At least, there
might be practical problems, but you could still pedal yourself up a
winch on a cable purely vertically perfectly well.

There's no fundamental reason why it would be better to walk if the
grade is 40% or steeper. Actually you'd be climbing on all fours at that
grade... but suppose we assume your bike is on a cable or funicular
railway and has low enough gears. Well, you might as well keep riding
it. Yes you do have to keep increasing your gravitational potential
energy, but that's the same cycling or walking.

Cycling in a low gear vs walking comes up on RBT quite often. Which is
more efficient comes down to whether the pedalling action or the walking
action is more efficient. I don't know how efficient the walking action
is, but the pedalling action is certainly highly efficient. Then you
have to consider whether you really care about efficiency-- lower
cadences are more energy efficient than higher but might make your legs
hurt more which matters more.

On Aug 1, 6:13*am, bicycle_disciple wrote:
Hi all. I would love some of the experienced math and physics gurus
here to review and critique my analysis of vertical climbing rate
(with application to cycling). If you have any insights or how I could
improve, that'd be great.http://tinyurl.com/nu2p5w

Yeah. I did beat it to death. Now to find out if I beat the right one
to death.

BD

It went off the rails at "Your speed will decrease exponentially and
at some critically steep grade possibly 40% or more, your velocity
will be reduced to near zero"

Which comes just after you had derived that climbing speed reaches a
nonzero asymptote w/r/t slope.

Try reconstructing your figure 2, calculate the lines for 20%, 40%,
100%, 200%, and infinite slope. Or plot the ground speed as constant
power as a function of slope, which (you will find) does not approach
zero. We can climb ladders after all.

"it is more practical energy wise to get off your bike and walk. Why?
Because the speeds are more or less the same cycling or walking!"

This presumes a relationship between "speed" and "efficiency" which
you have not explored. And both your page and Ferrari's use the term
"exponential increase" in a bizarre, broken way.

N.B. Any difference in efficiency between walking and cycling a slope
is going to come from biomechanics -- are the muscles contracting at
an efficient rate, what forces are the muscles generating over what
articulation of the joint -- and not from these energy considerations,
as all the analysis you present applies just as well to walking as to
cycling.

On Aug 1, 11:04*am, Ben C wrote:
On 2009-08-01, bicycle_disciple wrote:

Hi all. I would love some of the experienced math and physics gurus
here to review and critique my analysis of vertical climbing rate
(with application to cycling). If you have any insights or how I could
improve, that'd be great.http://tinyurl.com/nu2p5w

Yeah. I did beat it to death. Now to find out if I beat the right one
to death.

When you say Rise/Run, by "Run" you mean distance travelled along the
road as measured by your cycle's odometer for example? If so it looks
right.

In the section "Practical and Theoretical Limits of Climbing Rate", we
have a graph of climbing rate against grade for a road-speed of 1. So if
my road-speed is 1, and the grade is very steep-- practically vertical--
then my VAM is almost 1, which is why we approach that 1 line at the top
and bottom. Vertical speed equals road speed for a vertical road, and
can never be greater than road speed of course.

So far so good, but Note 2 is a bit strange. Grade can't keep increasing
without bounds-- it can never get steeper than 1. For a constant road
speed, then the steeper the grade the higher the VAM which may be what
Ferrari is saying.

Road grades are usually expressed as rise/run = vertical distance /
horizontal distance, so they can exceed 100%, and BD's introduction of
the sin(atan(rise/run)) correction would seem to indicate that is his
understanding of rise/run as well.

Whether you roll backwards or fall or your gears aren't low enough
doesn't really have anything to do with efficiency. At least, there
might be practical problems, but you could still pedal yourself up a
winch on a cable purely vertically perfectly well.

With appropriate gearing, it would probably be faster to pedal
yourself up a funicular than to hike the same slope dragging a
funicular-pedaling-device behind you.

-pm

pm wrote:

With appropriate gearing, it would probably be faster to pedal
yourself up a funicular than to hike the same slope dragging a
funicular-pedaling-device behind you.

On a really steep slope, a bicycle constitutes a stable walking stick
with a brake. That's worth something, in my opinion. Even if it's
just something to lean on while gasping for air.

Chalo

On 2009-08-01, pm wrote:
On Aug 1, 11:04*am, Ben C wrote:
On 2009-08-01, bicycle_disciple wrote:

Hi all. I would love some of the experienced math and physics gurus
here to review and critique my analysis of vertical climbing rate
(with application to cycling). If you have any insights or how I could
improve, that'd be great.http://tinyurl.com/nu2p5w

Yeah. I did beat it to death. Now to find out if I beat the right one
to death.

When you say Rise/Run, by "Run" you mean distance travelled along the
road as measured by your cycle's odometer for example? If so it looks
right.

In the section "Practical and Theoretical Limits of Climbing Rate", we
have a graph of climbing rate against grade for a road-speed of 1. So if
my road-speed is 1, and the grade is very steep-- practically vertical--
then my VAM is almost 1, which is why we approach that 1 line at the top
and bottom. Vertical speed equals road speed for a vertical road, and
can never be greater than road speed of course.

So far so good, but Note 2 is a bit strange. Grade can't keep increasing
without bounds-- it can never get steeper than 1. For a constant road
speed, then the steeper the grade the higher the VAM which may be what
Ferrari is saying.

Road grades are usually expressed as rise/run = vertical distance /
horizontal distance, so they can exceed 100%, and BD's introduction of
the sin(atan(rise/run)) correction would seem to indicate that is his
understanding of rise/run as well.

Yes, you're right. I was also trying to work backwards to what he meant
from the formulas but got it wrong.

We're looking for the vertical distance given the road-distance in a
given second, which is sin(theta), and theta is atan(rise/run), so
that's where sin(atan(rise/run)) comes from.

Whether you roll backwards or fall or your gears aren't low enough
doesn't really have anything to do with efficiency. At least, there
might be practical problems, but you could still pedal yourself up a
winch on a cable purely vertically perfectly well.

With appropriate gearing, it would probably be faster to pedal
yourself up a funicular than to hike the same slope dragging a
funicular-pedaling-device behind you.

I suspect so, yes.

On Sat, 01 Aug 2009 12:54:03 -0700, pm wrote:

snip


N.B. Any difference in efficiency between walking and cycling a slope is
going to come from biomechanics -- are the muscles contracting at an
efficient rate, what forces are the muscles generating over what
articulation of the joint -- and not from these energy considerations,
as all the analysis you present applies just as well to walking as to
cycling.

I can think of one important difference between cycling and walking. A
walker must expend energy to keep his body erect. If the cyclist is
seated, he is not. I'd expect that if both the cyclist and walker proceed
up the grade at the same speed, the cyclist would be expending less
energy.

Stephen Bauman

On 1 Aug, 19:04, Ben C wrote:
On 2009-08-01, bicycle_disciple wrote:

Hi all. I would love some of the experienced math and physics gurus
here to review and critique my analysis of vertical climbing rate
(with application to cycling). If you have any insights or how I could
improve, that'd be great.http://tinyurl.com/nu2p5w

Yeah. I did beat it to death. Now to find out if I beat the right one
to death.

When you say Rise/Run, by "Run" you mean distance travelled along the
road as measured by your cycle's odometer for example? If so it looks
right.

In the section "Practical and Theoretical Limits of Climbing Rate", we
have a graph of climbing rate against grade for a road-speed of 1. So if
my road-speed is 1, and the grade is very steep-- practically vertical--
then my VAM is almost 1, which is why we approach that 1 line at the top
and bottom. Vertical speed equals road speed for a vertical road, and
can never be greater than road speed of course.

So far so good, but Note 2 is a bit strange. Grade can't keep increasing
without bounds-- it can never get steeper than 1. For a constant road
speed, then the steeper the grade the higher the VAM which may be what
Ferrari is saying.

Whether you roll backwards or fall or your gears aren't low enough
doesn't really have anything to do with efficiency. At least, there
might be practical problems, but you could still pedal yourself up a
winch on a cable purely vertically perfectly well.

There's no fundamental reason why it would be better to walk if the
grade is 40% or steeper. Actually you'd be climbing on all fours at that
grade... but suppose we assume your bike is on a cable or funicular
railway and has low enough gears. Well, you might as well keep riding
it. Yes you do have to keep increasing your gravitational potential
energy, but that's the same cycling or walking.

Cycling in a low gear vs walking comes up on RBT quite often. Which is
more efficient comes down to whether the pedalling action or the walking
action is more efficient. I don't know how efficient the walking action
is, but the pedalling action is certainly highly efficient. Then you
have to consider whether you really care about efficiency-- lower
cadences are more energy efficient than higher but might make your legs
hurt more which matters more.

Fatigue sets the riders limitations and is the criteria on which he
should gear.

On 1 Aug, 21:15, Chalo wrote:
pm wrote:

With appropriate gearing, it would probably be faster to pedal
yourself up a funicular than to hike the same slope dragging a
funicular-pedaling-device behind you.

On a really steep slope, a bicycle constitutes a stable walking stick
with a brake. *That's worth something, in my opinion. *Even if it's
just something to lean on while gasping for air.


Or perch against.