An Olympic athlete set a world record of 9.33 s in the 100-m dash. Did his speed

Question

An Olympic athlete set a world record of 9.33 s in the 100-m dash. Did his speed ever exceed 38 km/hr during the race? Explain. Select the correct choice below and fill in any answer boxes to complete your choice. (Round to one decimal place as needed.) The average speed is km/hr. By the Mean Value Theorem, the speed was exactly km/hr at least once. By the Intermediate Value Theorem, all speeds between and km/hr were reached, therefore the athlete's speed never exceeded 38 km/hr. The average speed is km/hr. Since this value is below 38 km/hr, it is impossible to tell if his speed exceeded 38 km/hr. The average speed is km/hr. By the Mean Value Theorem, the speed was exactly km/hr at least once. By the Intermediate Value Theorem, all speeds between and km/hr were reached, therefore the athlete's speed exceeded 38 km/hr.

Explanation / Answer

V aver = distance over elapsed time = 100/9.33 = 10.718 m/sec
Vaver = 10.718m/sec = 10.718*3.6 = 38.5 km/h > 38 km/h
since the athlete starts from rest, is means that during the great part ( 80% approx.) of its acceleration the developped speed is below 38 km/h

If we assume he can accelerate at a rate of 5 m/sec^2 , then :
100 = 1/2*a*t^2+a*t*(9.33-t)
100 = 2.5*t^2+46.65*t-5t^2
100+2.5t^2-46.65t = 0
t = (46.65-46.65^2-1000)/5 = 2.47sec ...(acceleration time)

check
1/2*5*2.47^2+5*2.47*(9.33-2.55) = 100.00 m
Top speed = 5*2.47 = 12.35 m/sec

Vx = 38/3.6 = 10.56 m/sec (speed in m/sec corresponding to 38 km/h)
tx = Vx/a = 10.56/5 = 2.12 m/sec ( time required for getting 38 km/h with accel. of 5.0 m/sec^2)
this means that in the first 2.12 sec of the dash, the speed is constantly below 38 km/h ...for the rest of the time (9.33-2.12 = 7.21 sec) the speed will be higher than 38 km/h

The average speed was approx. 38.5km/hr. First, convert 100m to 0.1km. Divide that by 9.33s, so you get a value km/s. Multiply by 3600 and you get the his average speed per hour, which is 38.5km/hour

Since his average speed was 38.5km/hr, according to the mean value theorem which states that if the average rate of change is 39.5km/hr, then there is one point 'c' at which the instantaneous rate of change is 38.5km/hr. In other words, there is definitely a point in which he reached 38.5km/hour.

According to IVT, all speeds between 0 and 38.5km/hr were reached since there are values f'(c) = 0 for all values [a,b] with a being 0 and b being 38.5km/hr.

Yes, the speed of 38 km/hr was at some point exceeded, because the athlete's average speed was 38.5km/hr, and MVT states that at some point the athlete's average speed = instantaneous speed. Thus, at some point he rached 38.5km/hr. 38.5km/hr > 38km/hr.

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