Vectors- River Race. who won the race?

Question

a boy and a girl decide to have the "annual" who's the best contest. Both are capable of swimming at 4.000m/s. they decide to race from the south shore to a point directly north, on the opposite shore, 200.0m distant, across a river that is flowing 3.00m/s towards the east. the boy swims directly northward, and is pushed downstream, meaning that he has to run back to the finish line, at a speed of 7.500m/s. the girl heads into the current such that she ends up at the finish line. who won the race? By what margin?

Answer 1

The boy swims north at 4 m/s while being pushed east at 3 m/s.  It will take him 50 s to cross the 200 m river.  During this time he will have been pushed downstream 150 m.  It will take him 20 s to run the 150 m at 7.5 m/s.  So his total time will be 50 + 20 = 70 s. 

 

The girl swims at an angle upstream such that the vector sum of her velocity at 4 m/s and the current at 3 m/s east will result in a vector heading due north.  If you draw these vectors you will see they form a right triangle with the hypotenuse being her swim direction at 4 m/s, one leg being the current east at 3 m/s, and the other leg being her net velocity north.  Her net velocity north is sqrt(4^2 - 3^2) = sqrt(7) m/s.  It will take her 200/sqrt(7) = 75.593 s to swim across the 200 m river.  

 

So the boy will win the race by 5.593 s