Question 4.8: A Boat Crossing a River A boat crossing a wide river moves w...

(A) Conceptualize Imagine moving in a boat across a river while the current pushes you down the river. You will not be able to move directly across the river, but will end up downstream as suggested in Figure 4.22a. Imagine observer A on the shore, so that she is on the Earth, represented by letter E. Observer B is represented by letter r in the figure; this observer is on a cork floating in the river, at rest with respect to the water and carried along with the current. When the boat begins from point P and is aimed straight across the river, the velocities \overrightarrow{u}_{br}, the boat relative to the river, and \overrightarrow{v}_{rE}, the river relative to the Earth, add to give the velocity \overrightarrow{v}_{bE}, the velocity of the boat relative to observer A on the Earth. Compare the vector additionin Figure 4.22a to  that in Figure 4.21. As the boat moves, it will follow along vector \overrightarrow{v}_{bE}, as suggested by its position after some time in Figure 4.22a.

Categorize Because of the combined velocities of you relative to the river and the river relative to the Earth, we can categorize this problem as one involving relative velocities.

Analyze We know \overrightarrow{u}_{br}, the velocity of the boat relative to the river, and \overrightarrow{v}_{rE}, the velocity of the river relative to the Earth. What we must find is \overrightarrow{u}_{bE}, the velocity of the boat relative to the Earth. The relationship between these three quantities is \overrightarrow{u}_{bE}=\overrightarrow{u}_{br}+\overrightarrow{v}_{rE}. The terms in the equation must be manipulated as vector quantities; the vectors are shown in Figure 4.22a. The quantity \overrightarrow{u}_{br} is due north; \overrightarrow{v}_{rE} is due east; and the vector sum of the two, \overrightarrow{u}_{bE}, is at an angle θ as defined in Figure 4.22a.

Find the speed u_{bE} of the boat relative to the Earth using the Pythagorean theorem:

Find the direction of \overrightarrow{u}_{bE}:

Finalize The boat is moving at a speed of 11.2 km/h in the direction 26.6° east of north relative to the Earth. Notice that the speed of 11.2 km/h is faster than your boat speed of 10.0 km/h. The current velocity adds to yours to give you a higher speed. Notice in Figure 4.22a that your resultant velocity is at an angle to the direction straight across the river, so you will end up downstream, as we predicted.

(B) Conceptualize/Categorize This question is an extension of part (A), so we have already conceptualized and categorized the problem. In this case, however, we must aim the boat upstream so as to go straight across the river.

Analyze The analysis now involves the new triangle shown in Figure 4.22b. As in part (A), we know \overrightarrow{v}_{rE} and the magnitude of the vector \overrightarrow{u}_{br}, and we want \overrightarrow{u}_{bE} to be directed across the river. Notice the difference between the triangle in Figure 4.22a and the one in Figure 4.22b: the hypotenuse in Figure 4.22b is no longer \overrightarrow{u}_{bE}.

Use the Pythagorean theorem to find u_{bE} :

Find the direction in which the boat is heading:

Finalize The boat must head upstream so as to travel directly northward across the river. For the given situation, the boat must steer a course 30.0° west of north. For faster currents, the boat must be aimed upstream at larger angles.