A 0.20-km wide river has a uniform flow speed of 4.0 m/s toward the east. It takes 20 s for a boat to cross the river to a point directly north of its departure point on the south bank. In what direction must the boat be pointed in order to accomplish this?
A) 23° west of north
B) 20° west of north
C) 24° west of north
D) 22° west of north
E) 17° west of north

Question

Quizplus · Accepted Answer

First, we can use the given information to find the velocity of the boat relative to the water. Let Vb be the velocity of the boat and Vw be the velocity of the water:

Vb^2 + Vw^2 = (4.0 m/s)^2      (Pythagorean theorem)
Vw = 4.0 m/s east              (given)
Vb = sqrt((4.0 m/s)^2 - Vw^2) = 3.2 m/s north

So the boat must travel at 3.2 m/s north relative to the water. Now we can use some trigonometry to find the necessary angle. Let θ be the angle between the boat's velocity and due north:

tan(θ) = Vb/Vw
θ = tan^-1(Vb/Vw) = tan^-1(3.2/4.0) = 36.9°

However, this is the angle between the boat's velocity and due east (i.e. the direction of the river flow). To get the angle relative to due north, we need to subtract 90°:

θ' = 90° - θ = 53.1° west of north

So the boat must be pointed 53.1° west of north to travel directly north across the river. Rounded to the nearest degree, this is choice D: 22° west of north.

A 020-Km Wide River Has a Uniform Flow Speed of 4