A200-m-wide river has a uniform flow speed of 1.1 m/s through a jungle and toward the east. an explorer wishes to leave a small clearing on the south bank and cross the river in a powerboat that moves at a constant speed of 4.0 m/s with respect to the water. there is a clearing on the north bank 82 m upstream from a point directly opposite the clearing on the south bank. (a) in what direc- tion must the boat be pointed in order to travel in a straight line and land in the clearing on the north bank? (b) how long will the boat take to cross the river and land in the clearing?

angle 74º  and 21.3

Explanation:

In this problem we have to make a composition of velocity vectors to obtain a result, we see that the distance that the boat has to travel of 82 m, do not ask what direction it should take to travel directly to the opposite shore.

One of the easiest way to solve this problem is to use separate axes xy (East - North) and work them separately, let's build the speeds on each axis

  East

           Vrio + V boat = O

           V boatx = -V river = -1.1 m / s

The negative sign indicates that the boat must in the west, to meet the condition you must travel directly to the North

North

         V boaty = Vy

We have the speed of the boat 4.0 m / s, to meet the condition of traveling north the boat must point at a given angle, so the boat speed would be the tangent of the right triangle, from here we can find the angle by trigonometry

         cos θ = Vboatx / Vboat

         cos θ = -1.1 / 4.0 = 0.275

         θ = 106º

 This is the angle measured from the positive direction of the x-axis, to have the measurement in the second currant we subtract it from 180,

          180 -106 = 74º

In the second quadrant an angle of 74º North West

b) To find how long the trip takes, we use the Pythagorean theorem to find the vertical component of velocity, this case would be the northern component (opposite leg)

          V² = Vboatx² + Vboaty²

          Vboaty = √(V² -Vboatx²)

          Vboaty = √ (4² - 1.1²)

          Vboaty = 3.85 m / s

Having the effective speed to the north we can calculate the travel time

          V = x / t

          t = x / v

          t = 82 / 3.85

          t = 21.3 s

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