Barbara can swim at 4km/h in still water.

Question

She wishes to swim across a river to a point directly opposite from where she is standing.?The river is moving at a rate of 5km/h. Explain, with the use of a diagram, why this is not possible.
Isn't the bottom leg 4km/h and the other leg 5km/h?

Answer 1

If you position Barbara at the origin and orient the river so its left bank corresponds to the Y-axis and the current is flowing in the negative Y-direction then Barbara will be trying to swim in the positive X-direction to a point (W,0) where W is the width of the river.  

 

The current can be represented by a vector of length 5 km/h in the negative Y-direction, that is (0,-5).  The length of Barbara's swim vector is 4 km/h.  She can swim in any direction that has a positive X-component (because that's where the river is).  Her swim vector is represented by (x,y) where

 

x² + y² = 4  ==>  y² = 4 - x² ==>  y = ±√(4 - x²)  

 

This is any point on the half-circle going clock-wise from (0,4) to (0,-4).  So the Y-component of her swim vector must lie between -4 and 4.  

 

Barbara's total movement vector will be the vector sum of her swim vector and the current vector.  You can see that regardless of which direction she swims in, the addition of the current vector will make the Y-component of her total movement vector lie between -9 and -1, which means it is not possible for her to swim directly across the river.  In other words, she will be pushed downstream to some degree by the current regardless of which direction she swims in.