At the Beijing 2008 Olympic Games, Michael Phelps won his gold medal in 1:42.96

Question

At the Beijing 2008 Olympic Games, Michael Phelps won his gold medal in 1:42.96 in the Men's 200m Freestyle final. Tae-Hwan Park from the Republic of Korea was second in 1:44.85. If Park had foreseen the results and had decided to speed up by 20% at a certain point fo the race, he may have been able to win. Assume that Phelps held a constant speed throughout the race and Park held constant speeds both before and after the change in speed. Considering that increasing the speed requires a lot of energy, Park would have liked to change his speed as late as possible. When should park have made his move (in terms of distance)? (Please show all work)

Explanation / Answer

assume that speed of park is v. we have that v*104.85=L but park want to change it speed is 1.2*v in delta_T to match the result of Phelps 102.96. so that we have v*(102.96-deltaT)+1.2*v*deltaT=L so that v(102.96-deltaT)+1.2*v*deltaT=v*104.85 so that 102.96-deltaT+1.2*deltaT=104.85 so deltaT=9.45(s). because Park want this as late as possible so this must be the last 9.45sec of his race. so that at distace S=v*(102.96-9.45). park should starts to accelerate. so S=v*93.51=L*93.51/104.85 (because L/104.85=v. take a look above) so S=178.4(m)

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