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Chapter 3: Problem 27

You wish to row straight across a 63 -m-wide river. You can row at a steady \(1.3 \mathrm{m} / \mathrm{s}\) relative to the water, and the river flows at \(0.57 \mathrm{m} / \mathrm{s} .\) (a) What direction should you head? (b) How long will it take you to cross the river?

Short Answer

Expert verified
The rower should head \(24^\circ\) upstream to directly cross the river. It will take approximately \(48.46\) seconds to cross the river.

Step by step solution

01

Calculate the Resultant Velocity

The total velocity \(v_t\); which is the velocity that acts downstream due to the current of water, can be calculated using the Pythagoras theorem since we are dealing with a right-angled triangle scenario. Using the formula \(v_t = \sqrt{(v_r^2 + v_c^2)}\) where \(v_r\) is the rower's velocity(1.3 m/s) and \(v_c\) is the current's velocity(0.57 m/s), we find that \(v_t = \sqrt{(1.3^2 + 0.57^2)} = 1.41\) m/s.
02

Calculate the Direction Angle

We use the tangent function to find out the direction or angle θ the rowboat should be heading. So, \(\theta = \tan^{-1}(\frac{v_c}{v_r}) = \tan^{-1}(\frac{0.57}{1.3})\). Hence, θ is 24°. The rower should head 24° upstream.
03

Calculate the Time taken to cross the River

Time \(t\) taken to cross the river is given by the formula \(t = \frac{Distance}{Speed}\). The distance is the width of the river that is 63m and the speed would be the rower's speed as the rower's velocity is perpendicular to the river's current. So, \(t = \frac{63}{1.3} = 48.46\)s. It will take approximately 48.46 seconds to cross the river.

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