**Brian Beckman, PhD©Copyright August 2000**

In the previous instalment, we did exact calculations for a dummy line down a 650-foot entry straight, a 180-degree left-hander, and a 650-foot exit chute. Cornering radii vary from 150 feet to 200 feet, and the track is 100 feet wide all the way around. This dummy line carries constant speed around the entire left-hander. We did those calculations to provide reference times to compare against this month's more sophisticated calculations, in which we unwind the steering wheel and accelerate at the same time. The baseline times for the dummy line over the whole course, as a function of cornering radius, are in the second-to-last column of the following table:

Inscribed Corner Radius (ft) | Total time (sec) up to the apex | Time (sec) in corner after apex | Time for entrance and complete corner | Exit speed from chute (mph) @ g/2 accel | Time in exit chute (sec) | Combined segment time | Combined post-apex time and exit-chute time |
---|---|---|---|---|---|---|---|

150 | 11.872 | 0.000 | 11.872 | 109.091 | 5.670 | 17.541 | 5.670 |

152 | 10.912 | 0.860 | 11.773 | 107.857 | 5.528 | 17.301 | 6.388 |

154 | 10.544 | 1.209 | 11.754 | 107.422 | 5.460 | 17.213 | 6.669 |

155 | 10.401 | 1.348 | 11.750 | 107.260 | 5.430 | 17.180 | 6.779 |

160 | 9.872 | 1.881 | 11.753 | 106.697 | 5.308 | 17.061 | 7.189 |

170 | 9.208 | 2.600 | 11.808 | 106.101 | 5.116 | 16.924 | 7.716 |

180 | 8.762 | 3.126 | 11.888 | 105.806 | 4.955 | 16.844 | 8.082 |

190 | 8.424 | 3.556 | 11.980 | 105.666 | 4.813 | 16.792 | 8.369 |

200 | 8.150 | 3.927 | 12.077 | 105.627 | 4.682 | 16.760 | 8.609 |

From this point on, we need only look at the last column. It's after the apex
and down the exit chute where we look for improvement; we actually drive the
dummy line up to the apex. Many readers will be screaming that we *could*
try to get on the gas *before* the apex for even *more* improvement.
Others will be screaming "trail brake!," that is, ease off the brakes at the
same time as winding the steering wheel at turn in (thanks to reader Marc
Sibilia for pointing this out to me). We leave those refinements to later
articles.

The approach in this article is to find a line by building it up, step-by-step, honouring the traction circle and the sides of the track. This is one of the techniques we can use in computer simulations, so we get to kill two birds with one stone: previewing simulation and analysing a particular driving line. For convenience, we need a Cartesian coordinate system, that is, a square grid. Let's turn the track around 180 degrees for this purpose, and put the centre of the coordinate system at the centre of the corner. Since the inside edge of the track and the outside edge of the track are concentric semicircles, there is only one identifiable centre of the corner.

We'll work by measuring the position and heading of the centroid of the car
with respect to this new coordinate system. We have a goal of arriving at the
point *x* = 200, *y* = 650, measured in feet, in
the least possible time, with a heading of as close to 90 degrees as we can get
it, that is, heading straight down the track. We start at the apex, which
measures from *x* = *r*_{0} sin ,
*y* = *r*_{1} cos . The following sketch illustrates:

I must note, at this point, if you haven't already noticed, this instalment
of *The Physics of Racing* is going to be more concentrated and intense
than previous instalments. I'm just going to blurt out facts without the usual
explanations and walkthroughs. The reasons are (1) that we have a lot to get
through in a little space and (2) that we assume that if you've been following
the series this far, you've got the fortitude to work through it. So, *let's
get it on!*

The initial heading is tangent to the inner edge of the track, that is,
perpendicular to the line from the centre of the track's corner to the apex.
Therefore, it has the angleup from the horizontal *x* axis. We know the
starting speed, *v*0, so we know its components in the *x* direction
and in the *y* direction:
*v*0* _{x}* =

We perform the entire manoeuvre whilst never exceeding the limits of the
traction circle. We set those limits as 1*g* cornering and braking and
0.5*g* accelerating, with smooth transitions all way around, as in the
following sketch (the horizontal cap shows a way of accounting for engine
limitations with *non*-smooth transitions, which will allow us to
accelerate harder with the wheel still turned but probably scare us in the seat.
Also, we note that 0.5*g* is a plausible, if only approximate, number for
acceleration. We leave it to the reader to show that 0.5*g* in the quarter
mile results in a realistic 13-second elapsed time, if at an unrealistic speed
of 150 mph):

In each step of the calculation, we keep track of the following information:

- the time,
*t* - the current position,
*x*(*t*),*y*(*t*), which we check to make sure we're still on the track (*x*< 200) and to see whether we're done (y 650) - the current velocity,
*v*(_{x}*t*),*v*(_{y}*t*), which we use to update the current position: , and likewise for*y* - the tangential and radial acceleration,
*a*(_{t}*t*),*a*(_{r}*t*), that is, tangential and radial to the bit of racing line at each instant (the*instantaneous*line), which we check to make sure that we're not cornering over the limit and that we're not exceeding the capacity of the engine, i.e., that is inside the traction envelope - the acceleration in the
*x*and*y*directions,*a*(_{x}*t*),*a*(_{y}*t*), which we use to update the current velocity: , and likewise for*v*_{y}

We drive the whole simulation by feeding on the throttle linearly with time
over a time span called *k* and by simultaneously increasing the
instantaneous radius of the driving line over a potentially different time span
called *k*_{unwind}. Feeding on the throttle allows us to increase
the tangential acceleration, *a _{t}* at each time step, and
unwinding allows us to

Let's look at the first few rows of this simulation in a spreadsheet and delve into the formulas more deeply:

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

t |
a(t)
(tangential, fpsps) |
v/^{2}r
(radial, fpsps) |
a(t) (radial,
fpsps) |
r(t)
(feet) |
a(_{x}t)
(fpsps) |
a(_{y}t)
(fpsps) |
x(t)
(feet) |
y(t)
(feet) |
v(_{x}t)
(mph) |
v(_{y}t)
(mph) |
v (mph) |

0.00 | 0.00 | 32.00 | 32.00 | 160.00 | -21.33 | 23.85 | 66.67 | -74.54 | 36.36 | 32.52 | 48.79 |

0.20 | 1.28 | 31.90 | 30.27 | 169.92 | -21.20 | 21.64 | 76.80 | -64.41 | 33.46 | 35.66 | 48.90 |

0.40 | 2.56 | 31.59 | 28.54 | 182.30 | -20.76 | 19.75 | 86.09 | -53.42 | 30.59 | 38.51 | 49.18 |

0.60 | 3.84 | 31.06 | 26.81 | 197.64 | -20.06 | 18.19 | 94.54 | -41.64 | 27.79 | 41.12 | 49.63 |

0.80 | 5.12 | 30.32 | 25.08 | 216.59 | -19.17 | 16.96 | 102.20 | -29.13 | 25.10 | 43.54 | 50.25 |

0.90 | 5.76 | 29.85 | 24.22 | 227.68 | -18.67 | 16.47 | 105.74 | -22.62 | 23.80 | 44.69 | 50.63 |

1.00 | 6.40 | 29.33 | 23.35 | 240.01 | -18.13 | 16.05 | 109.09 | -15.94 | 22.53 | 45.80 | 51.04 |

[column 1]: increments by each row; we actually computed with = 0.05 sec and display here every fourth actual row; this is an independent column, meaning that it does not depend on data from any other column.

[column 2]: tangential acceleration,

,

accounting for squeezing on the throttle up to *g */ 2;
depends only on column 1.

[column 3]: maximal radial acceleration,

,

accounting for the traction circle; more precisely, for the upper half of the
circle treated as a flattened (*oblate*) ellipse with height
*g* / 2; depends only on column 2.

[column 4]: radial

,

accounting for unwinding the steering wheel; in steps from the inner
parentheses outwards:
*g*(1 - *t* / *k*_{unwind}) slowly
decreases from *g* as time increases from 0, but, it is never allowed to
exceed *v*^{2} / *r*, by the **min** expression,
as mandated by the traction circle, and then, never allowed to be negative, by
the **max** expression, because we don't want to start turning back toward
the entry straight; depends on columns 1 and 3.

[column 5]:

;

just for amusement, it's interesting to calculate the instantaneous radius of a circle we could be driving if we were not accelerating tangentially; depends on columns 4 and 12, but no other columns depend on this.

[column 6]:

,

this just selects out the *x* components of both the radial and
tangential accelerations, but makes sure that we never turn the wheel so much
that we start going to the left. Note that the radial acceleration *always*
tries to pull the car to the left, hence the minus sign (*centripetal*: see
part 4 of *The Physics of Racing*); depends on columns 2, 4, 10, 11, and
12.

[column 7]:

,

selecting the *y* components, this time always pointing down the track,
the way we want to go; depends on columns 2, 4, 10, 11, and 12.

[column 8]:

,

just update the *x* coordinate by the velocity from the prior time step;
depends on columns 8 (the prior row of itself) and 10.

[column 9]:

,

do likewise for the *y* coordinate; depends on columns 9 (prior row) and
11.

[column 10]:

,

for updating the *x* component of the velocity (but don't let it go
negative, checking yet again, and, yes, this is a *hack*); depends on
columns 10 (prior row) and 6.

[column 11]:

,

likewise for the *y* coordinate of the velocity; depends on columns 11
and 7.

[column 12]: finally,

,

depends on columns 10 and 11.

I've packed all this in an Excel spreadsheet. The spreadsheet should be in the download package for readers who acquired this document electronically.

Enough talk! Let's *drive!* Driving means playing with the values of
*r*, *k*, and *k*_{unwind}, and possibly even , to find the lowest overall time at which
columns 8 and 9 show 200 or less and 650 or more, respectively. In general,
"playing with" should be a sophisticated process involving hill climbing,
genetic search, simulated annealing, and other fancy strategies for finding the
very best values. In a computer simulation, we'd do that. However, we can do a
reasonable job, for the sake of demonstration, by just tweaking the numbers by
hand in the spreadsheet.

I have to admit that as I did so, I got kinaesthetic feelings as if I where
actually driving. When I 'ran off the track,' that is, picked numbers that gave
me *x* > 200, I gritted my teeth and blushed. When I was still
unwinding at the end, I got that panicky feeling of understeer, knowing that I
wasn't going to stay on after the end of the segment, and so on.

The best values I found by hand are shown in the following table at
*r* = 167.5, *k* = 3.25, and
*k*_{unwind} = 7.22. That means that we take 3.25 seconds
to bury the gas and 7.22 seconds to unwind the wheel. There are solutions with
lower segment times, but, since we're still unwinding long after the segment is
done, I reject these solutions as assuming too much about what's going on after
our segment is done. With more track to work with, however, *we can find lots
more time*. In fact, it's a slightly surprising fact that by taking 9 seconds
to unwind at *r* = 167.5, *k* = 3.25, we lose
hardly any time and stay 15 feet inside the outer edge. There is quite a bit of
territory to investigate even in this simple model.

r |
k |
k_{unwind} |
Besttime Found |
DummyTime |
Dummy-Best |
Best TotalTime Found |

155 | 1.500 | 2.000 | 6.500 | 6.779 | 0.279 | 16.901 |

160 | 2.500 | 3.700 | 6.875 | 7.189 | 0.314 | 16.747 |

165 | 3.000 | 5.950 | 7.050 | 7.482 | 0.432 | 16.550 |

167.5 | 3.250 | 7.22 | 7.120 | 7.605 | 0.485 | 16.466 |

170 | 3.500 | 8.550 | 7.225 | 7.716 | 0.491 | 16.433 |

175 | 4.000 | 11.170 | 7.400 | 7.912 | 0.512 | 16.367 |

180 | 4.500 | 13.330 | 7.575 | 8.082 | 0.507 | 16.337 |

185 | 5.000 | 30.000 | 7.700 | 8.233 | 0.533 | 16.282 |

Since the best dummy time, with the widest possible circle, is 16.760, and
the best time I found here was 16.466, **the improvement by unwinding and
accelerating simultaneously is 0.294 seconds**. This is very significant. If
the exit straight were longer, the improvement would be even more dramatic since
it would continue to accumulate time down the straight.

Note that this does *not* involve changing the entry to the corner other
than by slowing down! There is no trail braking or lifting-while-turning or
other risk-taking going on at corner entry. There is a very important driving
lesson, here: to go faster, it is not necessary to take risks on corner entry.
It is, in fact, * both safer and faster just to slow down on the
entry*. The improved exit will follow naturally from the combination of
looking far ahead and of being smooth. And that's not even fair!

There is no guarantee that this is the best possible improvement in the
model. I found these numbers by 'seat-of-the-pants' tweaking. A more systematic
or algorithmic search would very likely find better ones. In other words, I was
able to find almost three tenths by just driving a better line without trying
very hard at all. There is another driving lesson, here: * just driving a
better line gives better times time without changing the driver's margin for
error*, that is, without getting deeper into the

For the future, we can start taking more risks to get even more improvement. We can risk accelerating before the apex and we can risk deeper entry by trail braking, that is, easing off the brake and winding up the steering wheel at the same time. These makeovers do entail more driver risk since they are new opportunities for loss of car control.

**Erratum:** in part 17, I wrote "By driving a line just one foot larger
than the minimum, one is able to apex more than fifteen degrees later!". I
should have written "…fifteen degrees *earlier*!" The point was that the
tightest line does not apex until the geometric exit of the corner, and that's
*way too late*. The slip-of-the-pen occurred because one is so accustomed
to talking about late apexing as preferable.