And the Winner Is... Predicting the Outcome of the 150 Showdown

In a mere 9.84 seconds on July 27, 1996, Donovan Bailey performed multiple tasks at once: he won Olympic Gold, earned the title of "World's Fastest Man", defeated one of the best 100m final fields ever in record--breaking time, restored Canadian sprinting to world dominance, forever banished the tainted poltergeist of Ben Johnson, and shut out the United States from the medal positions. Meanwhile, in 19.32 seconds, Michael Johnson performed a similar number of tasks: he smashed beyond all recognition the 200m world record in a time faster than anyone though possible, he recorded blazing splits (10.12s, 9.20s), and (albeit due in part to the poor math skills of some sports commentators) his time evoked the thought in many minds that he was now the rightful heir to the title recently bestowed on Bailey.

Thanks to this hooplah, the Ottawa--based Magellan Group has set the stage for a match between these two one--man sprint powerhouses. On Sunday, June 1st at Skydome, almost 10 months after their respective world record performances, Donovan Bailey and Michael Johnson will speed over 150m in what is perhaps to be the most hyped Canada vs US sprint showdown since the 100m final in Seoul on September 24, 1988. The question on most minds is naturally: who will win?

There are numerous ways to guess the end result. One can consider the training of each individual. Bailey is a 100m specialist; does he have the power to generate world-class speed around a turn, then sustain it over a straight equal in length to his event? His best marks at 200m stand as 20.76s (20.39s wind-assisted) in 1994. Johnson, on the other hand, is a 200/400 doubler. Does he have the sheer power to match Bailey over half the race? His 100m PB is 10.09s, also from 1994. Surely, if nothing else, this problem almost borders on the philosophical! It has ringings of: "What happens when an irresistible force meets an immutable object?"

One can also look at the numbers that came out of Atlanta. Bailey was the last out of the blocks (reaction time of +0.174s), reached an astounding maximum velocity of 12.1 m/s at 60m, and accounting for tail-wind (+0.7 m/s), ran faster than did Ben Johnson at the Rome WC of 1987. Michael Johnson recorded the previously mentioned splits of 10.12s/9.20s, led the field at the line by over 0.3s in a sprint, and not counting his earlier 19.66s from the US Trials, broke the long--standing record of Pietro Mennea by 0.4s.

Statistician Robert Tibshirani of the University of Toronto performed such an analysis, taking into account a 1973 model of Mathematician Joseph Keller (who in turn based his model on the 1920s work of Physiologist A. Hill). It was found that, with a modification to Keller's model, Bailey would defeat Johnson by a margin of 0.03s-0.22s in 95 out of 100 races, or at a 95% confidence level (statistically-speaking).

However, Tibshirani noted tha his model did not take into account the energy lost by a sprinter (in this case, Bailey) as he rounds the curve. The Skydome track will have a 50m curve + 100m straight configuration, so such a model is limited by this non--linearity. What to do?

Contact a Physicist who has a love of track and field! Falling into such a category, I have tried my hand at prediction using a crystal ball of vector-addition. Without sounding too much like a Classical Mechanics class, let me briefly outline the problem and solution. The model used by Tibshirani assumes that the sprinter can exert a time-dependent force f(t) = f - ct; that is, as time progresses, he fatigues. If we know certain information about the race (e.g. splits, speeds), we can determine the values of f and c. However, the sprinter also feels internal resistances which sap his strength (anaerobic exhaustion, for example), so combining terms for all these effects, we can obtain a differential equation for his net acceleration.

Table 1: Predicted splits (s) and speed (m/s) compared with official for Bailey's 100 m final in Atlanta. Reaction time is rounded to +0.17 s.
Split	10 m	20 m	30 m	40 m	50 m	60 m	70 m	80 m	90 m	100 m
Speed (s)	9.32	10.95	11.67	11.99	12.10	12.10	11.99	11.85	11.67	11.47
Raw	1.89	2.90	3.79	4.64	5.47	6.29	7.12	7.96	8.81	9. 67
+reaction	2.06	3.07	3.96	4.81	5.64	6.46	7.29	8.13	8.98	9.84
Official	1.9	3.1	4.1	4.9	5.6	6.5	7.2	8.1	9.0	9.84

This works fine for a linear race (100m), but what of the turn? A sprinter of mass m rounding a turn of radius R feels a force of mv(t)²/R. Of course, in order to stay on the track (a useful strategy), he must combat this with his energy reserves. The centrifugal force mv(t)²/R acts perpendicular to the forward motion (dependent on the speed of the runner at time t), so we have to add the force terms as vector, the root of the sum of the squares f(t)² and m²v(t)⁴/R². So far so good, but such an assumption overcompensates for the curve, that is, it predicts 200m times which are well above what one would expect for world class performances. The key is to remember that a sprinter leans into the turn, letting gravity do some of the work combating the centrifugal force. So, a determination of exactly what fraction p of this force the runner actually feels can help predict times run on curves.

The 150m Showdown

Using our model, we can take a stab at how Bailey might be able to handle this race. The value of p we should use is still up in the air, but it seems likely that it won't be small. After all, Bailey is a 100m specialist, and wouldn't handle turns very well. So, we'll take p to be either 0.6, 0.7, or at the worst, 0.8 (these are the squares of the percentage of centrifugal forces felt, which would mean we're roughly considering between 75% and 90% of the force).

Michael Johnson's 50m split in his Olympic 200m final was about 6.4s, so assuming similar conditions, Bailey clearly leads off the curve. After this point, it's hard to determine how Johnson would handle the straight. His split of 10.12s was run on a complete turn, so we could guess that he could shave off about 0.05s. This would put his Skydome split at about 10.07s (slightly quicker than his 1994 PB). Continuing this logic, let's guess that he'd be able to hold a greater speed over the straight, and clock in between 0.05-0.10s faster than his Atlanta 150m split of 14.83s. Since Johnson is probably a more consistent curve runner, we could assume that his time doesn't vary more than 0.01s from lane to lane. If this is the case, then Johnson could optimistically clock between 14.73s-14.78s on June 1st.

The results of the model runs for Bailey are listed in Tables 2 and 3, and are broken up into 50m splits. Since the model calculates "raw" race times, a reaction time must be added on (I've assumed a +0.170s reaction, similar to Atlanta). The "final" race times are in the last column of the tables.

Table 2: Bailey's predicted Skydome 150m times for various values of p, assuming race is run in lane 3.
p	t50	t100	t150	t150+0.17
0.60	5.62	9.95	14.57	14.74
0.70	5.64	9.99	14.61	14.78
0.80	5.67	10.03	14.66	14.83

Table 3: Bailey's predicted Skydome 150m times for various values of p, assuming race is run in lane 4.
p	t50	t100	t150	t150+0.17
0.60	5.61	9.93	14.54	14.71
0.70	5.63	9.97	14.59	14.76
0.80	5.65	10.00	14.63	14.80

For the best Bailey guess (p=0.60), he takes Johnson regardless of lane choice (assuming 14.73s is an overestimate for Johnson). However, for larger p, which may be more realistic, Bailey's lane choice starts to become crucial. That is, for p=0.70, he can only win if he's assigned the outside lane. In the worst case, Bailey gets edged out, drained from fighting the curve.

How about 200m?

Tables 4 and 5 show predictions for outdoor and indoor races, respectively. Assuming the target range described above, then outdoor p values of 0.50-0.70 provide realistic estimates (20.28s-20.58s). Meanwhile, for indoor venues (Table~\ref{200indoors}), lower values of p give believable times. These are slower than the outdoor predictions, as one might expect, yet still within the grasp of the 100m champion (around 20.47s-20.86s).

Table 4: Bailey's predicted outdoor 200 m times, as run in lane 4.
p	t50	v50	t100	v100	t150	t200	t200 +0.16
0.25	5.53	11.74	9.89	11.03	14.56	19.81	19.97
0.36	5.55	11.60	9.98	10.85	14.69	19.96	20.12
0.50	5.59	11.43	10.09	10.65	14.84	20.13	20.29
0.60	5.61	11.31	10.16	10.51	14.93	20.24	20.40
0.70	5.63	11.20	10.24	10.39	15.09	20.43	20.59

Table 5: Bailey's predicted indoor 200 m times, as run in lane 4.
p	t50	t100	t150	t200	t200+0.16
0.20	5.62	9.91	14.88	20.32	20.48
0.30	5.68	10.01	15.17	20.71	20.87
0.40	5.75	10.13	15.43	21.05	21.21
0.50	5.81	10.22	15.67	21.37	21.53
0.60	5.88	10.32	15.91	21.68	21.84
0.70	5.94	10.42	16.13	21.97	22.13
0.80	5.99	10.50	16.33	22.23	22.39

The higher values of p for the indoor track give what are certainly inaccurate times, and hardly world class. Recall, though, that the centrifugal force depends on 1/R; a sprinter traveling at the same speed over a radius half as big feels twice the force. Since p is the square of the percentage of force felt, then a ratio of 4:1 for outdoor:indoor values of p seems reasonable. This is why tracks are banked, so the sprinter doesn't have to expend too much energy to compensate for the curve (unless the race is run in Sherbrooke, where the sprinter must fight not to fall into the center!).

And the winner is...

What will the victory signify? Should Johnson prevail, would he usurp Bailey's title of World's Fastest Man? In my opinion, no. Bailey won the traditional event to claim the title, set a world record, and achieved a higher speed than Johnson. American sour-grapes aside, this spectacle will only serve to show who would win over 150m. And truthfully, the winners will be Bailey and Johnson, who will walk from Skydome a combined $1.5M richer than they were the day before. Canadian Track and Field will win, because the event will hopefully regenerate significant interest in the sport. Finally, the audience will win, because they will be treated to a magnificent race between two of history's greatest sprinters.

Addendum

Using the values p=0.60, 0.70, and 0.80, the corresponding race times are presented below (Table 6). It is clear from this that, according to this model, Bailey's times will be almost 0.1s slower, simply from running an extra 25m curve! In this case, the curve radius has been chosen as being roughly that of lane 8 of an outdoor track. If the radius is larger, then the times could conceivably come down.

Table 6: Bailey's predicted Skydome 150m times for a track configuration 75m curve + 75m straight (radius of curvature equivalent to lane 8 of outdoor track).
p	t50	t100	t150	t150+0.17
0.60	7.78	10.00	14.65	14.82
0.70	7.82	10.05	14.71	14.88
0.80	7.85	10.10	14.76	14.92

While Bailey will have the psychological advantage of the inside lane (that is, he will be able to push himself to catch Johnson), he will have the physical handicap associated therein. From Bailey's 100m WR race, it would seem that his strength is surging from behind, since he was last out of the blocks there. Meanwhile, Johnson's strength is running the curve, and the choice of a longer curve gives him a distinct advantage.

The odds on Johnson winning are rising to 3 to 1 in Las Vegas. Has the Magellan Group unknowingly given the $1M purse to Johnson? Unfortunately, the physics of the situation would tend to suggest `yes': Bailey loses too much energy around the longer curve, despite the larger radius. But, both sprinters have a way of surprising the predictions. Only the events of June 1st will truly decide the victor!