Math Conundrum regarding Usain Bolt's 100m world record

Question

Consider the equation, $S = ut + 1/2 at^2 $

Usain bolt ran $100$ meters in $ 9.58$ seconds for the world record, and going by the equation above, his acceleration over a distance of 100 meters will be as under $(u=0)$:

$S= 1/2 at^2 \to; a = S*2/t^2$, which gives $a= 100*2/(9.58)^2 = 2.18\, \text{m/s}^2 $

If acceleration is $2.18\, \text{m/s}^2$, Usain Bolt's final velocity after 9.58 seconds works out as $2.18*9.58 = 20.88\, \text{m/s} (75\,\text{km/h})$ - which is way above the maximum velocity he is supposed to have achieved $(12.44\,\text{m/s})$.

Where is the catch?

Answer 1

The catch is simple: You calculations are (about) correct, with a lot of ugly rounding and so on but that's not the point. Since the result is inconsistent with reality, the model $$s(t) = \frac a2 t^2$$ must be wrong for his run. (i.e. he did not run with constant acceleration)

Answer 2

Here is the solution:

Usain Bolt ran the 100 m race in 2 parts (mathematically) as under:.

Constant acceleration for 3.08 seconds: from start (u=0) to achieving final velocity of 12.44 m/sec, and
Constant velocity for 6.50 seconds: after reaching the final velocity of 12.44 m/sec.

thereby giving a total time of 3.08+6.50 = 9.58 seconds.

The calculations can be verified by the following equation:

Stotal = Stca + Stcv --------------------equation UB100

where,

Stotal = Total distance covered

Stca = Distance covered under constant acceleration

Stcv = Distance covered under constant velocity.

Stotal = 100 m

Stca =1/2 a(tvf)2 = 1/2 a * (3.08)2 = 4.74a m

Stcv = vf * (tf-tvf) = 12.44 * (9.58-3.08) = 80.86 m

where,

vf = final velocity (12.44 m/sec)

tvf = time to reach final velocity (3.08 sec)

tf = time to finish the 100m race (9.58 sec)

a = constant acceleration (?)

Substituting values of Stotal, Stca and Stcv in equation UB100, we get:

100 = 4.74a + 80.86

4.74a = 100 – 80.86

4.74a = 19.14

a = 4.038 m/sec2

Final velocity can thus be calculated as under:-

vf = ut0 + atvf

vf = 0 + 4.038 * 3.08 = 12.44 m/sec