In a harbour, wind is blowing at the speed of 72 km/h and the flag on the mast of a boat anchored in the harbour flutters along the N-E direction. IF the boat starts moving at a speed of 51 km/h to the north, what is the direction f the flag on the mast of the boat ?

Explanation

When the boat anchored in the harbour, the flag flutters along the N-E direction. It shows that the velocity of wind is along the North-East direction. When the boat starts moving, the flag will flutters along direction of relative velocity of wind w.r.t. boat. Let `vec(v)_(w b)` be the relative velocity of wind w.r.t. boat and `beta` be the angle between `vec(v)_(w b)` and `vec(v)_(w)`. Now
`vec(v)_(wb) = vec(v)_(w) + (-vec(v)_(b))`
Here `|vec(v)_(w)| = 72` km/hr
`vec(v)_(b) = 51` km/h
Angle between `vec(v)_(w)` and `-vec(v)_(b)` is `135^(@)` i.e. `theta = 135^(@)`, then

`tan beta = (51 sin 135^(@))/(72 + 51 cos 135^(@))`
`= (51 sin 45^(@))/(72 + 51(-cos 45^(@)))`
`= (51 xx 1//sqrt(2))/(72 - 51(1//sqrt(2))) = 1.0039`
`therefore beta = tan^(-1) (1.0039) = 45.1^(@)`
Angle w.r.t. each direction `= 45.1^(@) - 45^(@) = 0.1^(@)`
It means the flag will flutter almost due east.