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Chapter 3: Problem 75

A dolphin wants to swim directly back to its home bay, which is $0.80 \mathrm{km}\( due west. It can swim at a speed of \)4.00 \mathrm{m} / \mathrm{s}$ relative to the water, but a uniform water current flows with speed \(2.83 \mathrm{m} / \mathrm{s}\) in the southeast direction. (a) What direction should the dolphin head? (b) How long does it take the dolphin to swim the \(0.80-\mathrm{km}\) distance home?

Short Answer

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Problem: A dolphin wants to reach its home 0.80 km away to the west while swimming at a speed of 4.00 m/s relative to the water. A water current flows with a speed of 2.83 m/s in the southeast direction. Find the direction in which the dolphin should swim and the time it will take to reach its home. Solution: The dolphin should swim exactly west, heading at 180 degrees. It will take 400 seconds for the dolphin to reach its home.

Step by step solution

01

Understand the problem

First, let's analyze the given information. The dolphin wants to reach its home 0.80 km away to the west, and it swims at a speed of 4.00 m/s relative to the water. Meanwhile, a water current flows with a speed of 2.83 m/s in the southeast direction. The goal is to find the direction in which the dolphin should head and the time it will take to reach its home.
02

Break down the direction problem into vectors components

To determine the direction the dolphin should head, we will work with vectors. The water current has a southeast direction, so we can break down this velocity vector into its east and south components. Using the notation \(\text{v}_\text{C}\) for the velocity of the current, let \(v_\text{CE}\) be the east component and \(v_\text{CS}\) be the south component. Since the current flows at a 45-degree angle to the southeast, we have: \(v_\text{CE} = -v_\text{CS} = -2.83\; \mathrm{m/s} \times \frac{1}{\sqrt{2}} \approx -2.00\; \mathrm{m/s}\)
03

Calculate the velocity of the dolphin with respect to the ground

Now, let's find the velocity of the dolphin relative to the ground. Let \(\text{v}_\text{DG} = (\text{v}_\text{DE}, \text{v}_\text{DS})\) be the components of the velocity of the dolphin with respect to the ground and \(\text{v}_\text{D}\) be the velocity of the dolphin relative to the water. Since \(\text{v}_\text{DG} = \text{v}_\text{D} + \text{v}_\text{C}\), we have: \(v_\text{DE} = v_\text{D} \cos{\theta} - 2.00\; \mathrm{m/s}\) \(v_\text{DS} = v_\text{D} \sin{\theta} + 0\)
04

Apply the velocity equation to find the direction

To reach its home bay, the dolphin should move only to the west without any vertical movement. This means that \(v_\text{DS} = 0\). From the equation in the previous step, we have: \(v_\text{D} \sin{\theta} = 0\) Thus, \(\theta = 0^\circ\) or \(\theta = 180^\circ\). Now, since the dolphin swims at a speed of 4.00 m/s, we know its horizontal component of velocity, \(v_\text{DE}\), is smaller than 4.00 m/s. So the minus sign should be in front of the cosine term, which means \(\theta\) should be \(180^\circ\). Therefore, the dolphin must head exactly west.
05

Find the time it takes for the dolphin to reach its home

To find the time it takes for the dolphin to swim the 0.80 km distance, we will use the distance formula: \(t = \frac{d}{v_\text{DE}}\) Since \(d = 0.80\;\mathrm{km} = 800\;\mathrm{m}\) and \(v_\text{DE} = 4.00\; \mathrm{m/s} \cos{180^\circ} - 2.00\; \mathrm{m/s} = -2.00\; \mathrm{m/s}\), we obtain: \(t = \frac{800\;\mathrm{m}}{-2.00\;\mathrm{m/s}} = 400\;\mathrm{s}\) Thus, it takes the dolphin 400 seconds to swim the 0.80 km distance to its home.

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