A man swims across a river with speed of ` 5 km h^(-1)` ( in still water). While a boat goes upstream with speed ` 12 km h^(-1)` ( in still water). How fast and in which direction does, the man appear to go to the boatman ? Given that the speed of flowing water is ` 2 km h^(-1)`

Explanation

speed of flowing water ` v_(w) = 2 " km h"^(-1)`
speed of boat in still water ` , v_(b) = 12 " km h"^(-1)`
Boat goes upstrem, so that net speed of boat with respect to a peson standing on ground is ` | vecV_(bn)|= 12-2`
` 10 km h^(-1)`
According to the choice fo x-y axes as shown in figure. `vecv_(bn) = -190 hati`
Similarly the velocity of man is still water ` vecv_(m) = 5 hatj`
The net velocity of man in flowing water = `vecv_(mn) = vecv_(m) + vecv_(w) = 5hatj + 2 hat i `
` therefore vecv_(mn) = 2hati + 5hatj`

According to the defination of relative velocity.
the velocity of man w.r.t the botmari
` vecv_(mb) = vecv_(mn) = 2hati + 5hatj - (-10hati)`
` vecv_(mb) = 12hati + 5hatj`
magnitude of ` vecv_(mb) = sqrt((12)^(2) + 5^(2))`
` = sqrt(144 + 25)`
` v_(mb) = 12 " km h"^(-1)`
Direction of `vecv_(mb), theta = tan^(-1) ((V_(mby))/(V_(mby))) " " [ V_(mbx)= 12, V_(mby) = 5]`
` Rightarrow theta = tan^(-1) ( 5/12) = 22.6^(@)` with positive x-axis .
Thus to the boatman, the man appears to swin at a speed ` 13 km h^(-1)` in a direction making angle ` 22.6^(@)` with x-axis.