Naively, one could compute Johnson’s time in the 100m by first calculating his average speed in the 40,
speed = (distance) / (time)
= (40 yd) / (4.24 s)
= 8.6 m/s.
Using this speed, we could then calculate his time in the 100m,
time = (distance) / (speed)
= (100 m) / (8.6 m/s)
= 11.6 s.
That’s barely a decent time for a high school student, never mind the fastest man in the NFL. Bolt would blow him away if this were his actual speed. Where did we go wrong? The problem is that this is only an upper bound on Johnson’s time. The reason is that we calculated his speed using his 40 time, and much of the first 40 yards of a sprint are spent accelerating from a low speed. If we assume Johnson accelerates uniformly for the whole run, we could calculate his acceleration,
acceleration = 2 · (distance) / (time)2
= 2 · (40 yd) / (4.24 s)2
= 4.07 m/s2.
If we then assume he accelerates uniformly throughout 100m, we can calculate his new time,
t = 2 · (distance) / (acceleration)
= 2 · (100 m) / (4.07 m/s2)
= 7.01 s.
= 2 · (100 m) / (4.07 m/s2)
= 7.01 s.
That’s better than even the most unrealistic tricked out Madden player you can make. The reason is simple: since we assumed he was accelerating throughout the whole race, he has a much higher final speed then he would in reality. How much higher? About 64 mph!
Realistically, Johnson would probably reach his top speed somewhere between 10 and 40 yds and then maintain a roughly constant speed throughout the rest of the race. This being the case, he would almost certainly finish somewhere between these two bounds. Logic would seem to indicate that Bolt, typically regarded as the fastest man in the world, would still win handily. Unfortunately, our simple estimation is not precise enough to say who would win with any certainty. This is an important point. Sometimes we don't have enough background information to calculate definitive answers, and we can only calculate upper and lower bounds. Sadly, we'll have to deal with ESPN pundits endlessly debating who the faster man is.
Realistically, Johnson would probably reach his top speed somewhere between 10 and 40 yds and then maintain a roughly constant speed throughout the rest of the race. This being the case, he would almost certainly finish somewhere between these two bounds. Logic would seem to indicate that Bolt, typically regarded as the fastest man in the world, would still win handily. Unfortunately, our simple estimation is not precise enough to say who would win with any certainty. This is an important point. Sometimes we don't have enough background information to calculate definitive answers, and we can only calculate upper and lower bounds. Sadly, we'll have to deal with ESPN pundits endlessly debating who the faster man is.
Actually, you could get a better approximation by looking at the acceleration rates of all the sprinters in the Beijing 2008 Olympic final where Bolt set the current WR 9.58s. Split times are given for each 20m increment with a 30m split time. The Science of Sport Blog has done some analysis of those times here:
ReplyDeletehttp://www.sportsscientists.com/2009/08/analysis-of-bolts-958-wr.html
For the record Bolt would destroy Johnson in a 100m race. The 40 yd dash is measured off the first step of the athlete therefore eliminating reaction time as a factor. No time to figure it out at the moment, but I'd estimate Johnson at around a 10.12 - 100m runner.