Google Groups no longer supports new Usenet posts or subscriptions. Historical content remains viewable.
Dismiss
Effect of lane position in a blind left turn
20 views
Skip to first unread message
Joe Riel
May 22, 2013, 2:33:40 AM5/22/13
to
Just trying to introduce a bit of tech into the discussion.
Consider a worst-case scenario roadway; a dense forest
immediately adjacent to the edge of the road so sightlines
are blocked in curves. Let
w = width of each lane
R = radius of turn (at centerline of road)
d = distance of driver from centerline of road
b = distance of bicyclist from edge of lane
Vc = velocity of car
Vb = velocity of bike
kv = Vb/Vc
g = gravitational acceleration
mu = coefficient of friction
In a left-hand turn (driver and cyclist are traveling in the same
direction, on the right side of the roadway), the straight-line
distance at which the driver can just see the cyclist is
l = sqrt((R+d)^2 - (R-w)^2) + sqrt((R+w-b)^2 - (R-w)^2)
We are interested in the change of this distance as the
cyclist moves away from the edge (as b increase from 0
to w). To first-order, for w/R << 1, this is given by
dl/db*(l/w) ~= -1/2/(2+sqrt(2)) ~= -0.15
That is, if the bicyclist moves completely across the lane
to the center-line, the distance decreases by 15%. Moving
halfway across the lane, that is, to the center of the lane,
decreases the distance by half that, or 7.5%.
Assume car is braking with a constant deceleration, a.
To avoid hitting the cyclist the car has to slow from
vc to vb. This takes time
t = (Vc-Vb)/a.
During this time the distance to the cyclist decreases by
s = 1/2/a*(Vc-Vb)^2
Assume the car is traveling at maximum cornering velocity for curve,
which is approximately
Vc = sqrt(mu*g*(R+w/2))
Further assume a = mu*g (these are reasonable ballpark assumptions).
Ignoring the driver's reaction time for now, solve s = l with
d=0 (driver at centerline), b=0, kv=0, cyclist stopped at right
edge of roadway:
R ~= 45*w
For a 10ft wide lane, the radius of the centerline can be a maximum
of 450 feet and the car will be able to stop without hitting the
cyclist.
If the stopped cyclist moved to the center line, this distance is
R ~= 31*w
Now the maximum radius is 310 feet. These are very gentle turns;
reducing the radius improves the situtation because the car has
to slow due to the limit of adhesion.
--
Joe Riel
Consider a worst-case scenario roadway; a dense forest
immediately adjacent to the edge of the road so sightlines
are blocked in curves. Let
w = width of each lane
R = radius of turn (at centerline of road)
d = distance of driver from centerline of road
b = distance of bicyclist from edge of lane
Vc = velocity of car
Vb = velocity of bike
kv = Vb/Vc
g = gravitational acceleration
mu = coefficient of friction
In a left-hand turn (driver and cyclist are traveling in the same
direction, on the right side of the roadway), the straight-line
distance at which the driver can just see the cyclist is
l = sqrt((R+d)^2 - (R-w)^2) + sqrt((R+w-b)^2 - (R-w)^2)
We are interested in the change of this distance as the
cyclist moves away from the edge (as b increase from 0
to w). To first-order, for w/R << 1, this is given by
dl/db*(l/w) ~= -1/2/(2+sqrt(2)) ~= -0.15
That is, if the bicyclist moves completely across the lane
to the center-line, the distance decreases by 15%. Moving
halfway across the lane, that is, to the center of the lane,
decreases the distance by half that, or 7.5%.
Assume car is braking with a constant deceleration, a.
To avoid hitting the cyclist the car has to slow from
vc to vb. This takes time
t = (Vc-Vb)/a.
During this time the distance to the cyclist decreases by
s = 1/2/a*(Vc-Vb)^2
Assume the car is traveling at maximum cornering velocity for curve,
which is approximately
Vc = sqrt(mu*g*(R+w/2))
Further assume a = mu*g (these are reasonable ballpark assumptions).
Ignoring the driver's reaction time for now, solve s = l with
d=0 (driver at centerline), b=0, kv=0, cyclist stopped at right
edge of roadway:
R ~= 45*w
For a 10ft wide lane, the radius of the centerline can be a maximum
of 450 feet and the car will be able to stop without hitting the
cyclist.
If the stopped cyclist moved to the center line, this distance is
R ~= 31*w
Now the maximum radius is 310 feet. These are very gentle turns;
reducing the radius improves the situtation because the car has
to slow due to the limit of adhesion.
--
Joe Riel
Andre Jute
May 22, 2013, 12:05:33 PM5/22/13
to
Excellent analysis Joe. Saved for permanent reference. Thanks.
Andre Jute
Andre Jute
James
May 22, 2013, 6:12:52 PM5/22/13
to
brake and not hit the bike rider?
> Further assume a = mu*g (these are reasonable ballpark assumptions).
> Ignoring the driver's reaction time for now, solve s = l with
> d=0 (driver at centerline), b=0, kv=0, cyclist stopped at right
> edge of roadway:
>
> R ~= 45*w
>
> For a 10ft wide lane, the radius of the centerline can be a maximum
> of 450 feet and the car will be able to stop without hitting the
> cyclist.
>
> If the stopped cyclist moved to the center line, this distance is
>
> R ~= 31*w
>
> Now the maximum radius is 310 feet. These are very gentle turns;
> reducing the radius improves the situtation because the car has
> to slow due to the limit of adhesion.
>
--
JS
Joe Riel
May 22, 2013, 6:54:59 PM5/22/13
to
at the limit of adhesion is a maniac and won't be hitting the binders.
Yeah, that may the case, but this is a worst-case scenario.
>
>> Further assume a = mu*g (these are reasonable ballpark assumptions).
>> Ignoring the driver's reaction time for now, solve s = l with
>> d=0 (driver at centerline), b=0, kv=0, cyclist stopped at right
>> edge of roadway:
>>
>> R ~= 45*w
>>
>> For a 10ft wide lane, the radius of the centerline can be a maximum
>> of 450 feet and the car will be able to stop without hitting the
>> cyclist.
>>
>> If the stopped cyclist moved to the center line, this distance is
>>
>> R ~= 31*w
>>
>> Now the maximum radius is 310 feet. These are very gentle turns;
>> reducing the radius improves the situtation because the car has
>> to slow due to the limit of adhesion.
>>
>
> A 100 yard radius seems very big to me. How fast was the car traveling?
I made no assumptions about speed. But if he was pulling 0.7g
>> Further assume a = mu*g (these are reasonable ballpark assumptions).
>> Ignoring the driver's reaction time for now, solve s = l with
>> d=0 (driver at centerline), b=0, kv=0, cyclist stopped at right
>> edge of roadway:
>>
>> R ~= 45*w
>>
>> For a 10ft wide lane, the radius of the centerline can be a maximum
>> of 450 feet and the car will be able to stop without hitting the
>> cyclist.
>>
>> If the stopped cyclist moved to the center line, this distance is
>>
>> R ~= 31*w
>>
>> Now the maximum radius is 310 feet. These are very gentle turns;
>> reducing the radius improves the situtation because the car has
>> to slow due to the limit of adhesion.
>>
>
> A 100 yard radius seems very big to me. How fast was the car traveling?
that would be 55 mph (sqrt(0.7*300ft*32ft/s^2)/1.5.
The g-force assumed for a turning-speed advisory sign
is, I believe, around 0.3g. That corresponds to 35 mph.
--
Joe Riel
0 new messages