A boat travels 15 kph in still water. If it is headed degree 30 degrees North to West in a current that moves at 5 kph due west, what is the resultant velocity of the boat?
The speed of the boat in still water is 15 km/h. It is headed in a direction 30 degrees north of west. The speed of the river's current is 5 km/h and its direction is towards the west.
The resultant velocity of the boat is the sum of the velocity of the boat and the velocity of the current.
As the river is headed 30 degrees north of west, the component of its velocity towards the west is 15*cos 30 = `7.5*sqrt 3` and the component of its velocity towards the north is 7.5. The stream flows at 5 km/h due west. The two velocities due west can be added to give the component of the resultant velocity in the western direction as `7.5*sqrt 3 + 5` and the component of the resultant velocity towards the north is 7.5. The magnitude of the resultant velocity is `sqrt(7.5^2 + (7.5*sqrt3+5)^2) = 19.49` km/h. The direction of the velocity is at an angle of `tan^-1(7.5/(7.5*sqrt3 + 5)) = 22.63 ` degrees north of west.
The resultant velocity of the boat has a magnitude of` `approximately 19.49 km/h and the direction is 22.63 degrees north of west.