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)= 2kbarm h^(-1)`
Speed of boat in still water `v_(b) = 12 km h^(-1)`
Boat goes upstrem . So the net speed of boat with respect to a person standing on ground is `|barv_(bn)|= 12-2 `
=` 10 km h^(-1)`
According to the choice of x-y axes as shown in figure
`barv_(bn)=-10 hai`
Similarly the velocity of man in still water `barv_(m)=5hatj`
The net velocity of man in flowing water `barv_(mn)= bar(v)_(m)+barv_(w)`
= `5 hatj+2hati`
`:. barv_(mn)= 2hati+5hatj`
According to the definition of relative velocity .
The velocity of man w.r.t . the boatman
`barv_(mb)= bar(mn)-barv_(n)`
`=2hati+5hatj-(-10hati)`
`vec(v)_(mb)=12hati+5hatj`
`:.` Magnitude of `vec(v)_(mb)=sqrt((12)^(2)+5^(2))=sqrt(144+25)`
`V_(mb)=13 kmh^(-1)`
Direction of `barv_(mb) theta=tan^(-1)((V_(mb_(y)))/(V_(mb_(x))))[(v_(mb_(x))=12),(v_(mb_(y)=5))]`
`implies theta = tan^(-1) ((5)/(12)) ` with positive x-axis .
Thus to the man appears to swim at a speed 13 km `h^(-1)` in a direction making angle `tan^(-1)((5)/(12))` with x-axis .