A river flows due east at 3.0 m/s.A boat crosses the 300-m-wide river by maintaining a constant velocity of 10 m/s due north relative to the water.If no correction is made for the current,how far downstream does the boat move by the time it reaches the far shore?
A)6 m
B)30 m
C)60 m
D)90 m

Question

Quizplus · Accepted Answer

The boat's motion can be broken down into two components: its velocity due north (10 m/s) and the river's velocity due east (3.0 m/s). The boat's actual velocity relative to the shore is the vector sum of these two velocities. Using the Pythagorean theorem, we can find the magnitude of this velocity to be `sqrt(10^2 + 3^2) = sqrt(109)`.

The angle between the boat's velocity and the river's velocity is given by `tan(theta) = 3/10`, which implies `theta = 17.98 degrees`.

The boat takes `t = 300/10 = 30 s` to cross the river. In this time, the boat will have traveled a horizontal distance equal to its velocity relative to the shore multiplied by the time, or `10 * 30 = 300 m`. However, due to the river's current, the boat will also have drifted downstream. The horizontal distance traveled downstream is given by `d = vt = 3 * 30 = 90 m`. Therefore, the boat moves `90 m` downstream by the time it reaches the far shore.

A River Flows Due East at 3