Question

for questions 1 and 2: consider a boat which begins at point a and heads straight across a 100 - meter wide river with a speed of 8 m/s (relative to the water). the river water flows south at a speed of 3 m/s (relative to the shore). the boat reaches the opposite shore at point c. 1. which of the following would cause the boat to reach the opposite shore in more time? circle all that apply. justify each answer choice. a. the river is 80 meters wide. b. the river is 120 meters wide. c. the boat heads across the river at 6 m/s. d. the boat heads across the river at 10 m/s. e. the river flows south at 2 m/s. f. the river flows south at 4 m/s. g. nonsense! none of these effect the time to cross the river.

Explanation:

Step1: Recall time - distance - speed formula

The time \(t\) taken by the boat to cross the river is given by \(t=\frac{d}{v_{\perp}}\), where \(d\) is the width of the river and \(v_{\perp}\) is the component of the boat's velocity perpendicular to the flow of the river. The velocity of the river flow does not affect the time taken to cross the river because the river - flow velocity is parallel to the banks of the river and the motion of the boat across the river is perpendicular to the river - flow direction.

Step2: Analyze option a

If \(d = 80\) m (original \(d = 100\) m), using \(t=\frac{d}{v_{\perp}}\), with \(v_{\perp}\) constant, a smaller \(d\) gives a smaller \(t\). So this option will cause the boat to take less time to cross.

Step3: Analyze option b

If \(d = 120\) m (original \(d = 100\) m), using \(t=\frac{d}{v_{\perp}}\), with \(v_{\perp}\) constant, a larger \(d\) gives a larger \(t\). So this option will cause the boat to take more time to cross.

Step4: Analyze option c

If \(v_{\perp}=6\) m/s (original \(v_{\perp}=8\) m/s), using \(t = \frac{d}{v_{\perp}}\), with \(d\) constant, a smaller \(v_{\perp}\) gives a larger \(t\). So this option will cause the boat to take more time to cross.

Step5: Analyze option d

If \(v_{\perp}=10\) m/s (original \(v_{\perp}=8\) m/s), using \(t=\frac{d}{v_{\perp}}\), with \(d\) constant, a larger \(v_{\perp}\) gives a smaller \(t\). So this option will cause the boat to take less time to cross.

Step6: Analyze option e and f

The velocity of the river flow (\(v_{river}\)) is parallel to the banks of the river. Since the motion of the boat across the river is perpendicular to the river - flow direction, changing \(v_{river}\) from \(3\) m/s to \(2\) m/s (option e) or \(4\) m/s (option f) does not affect the time taken to cross the river.

Answer:

b. The river is 120 meters wide.
c. The boat heads across the river at 6 m/s.