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0. Math Review31m
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1. Intro to Physics Units1h 29m
2. 1D Motion / Kinematics3h 56m
3. Vectors2h 43m
4. 2D Kinematics1h 42m
5. Projectile Motion3h 6m
6. Intro to Forces (Dynamics)3h 22m
7. Friction, Inclines, Systems2h 44m
8. Centripetal Forces & Gravitation7h 26m
9. Work & Energy1h 59m
10. Conservation of Energy2h 54m
11. Momentum & Impulse3h 40m
12. Rotational Kinematics2h 59m
13. Rotational Inertia & Energy7h 4m
14. Torque & Rotational Dynamics2h 5m
15. Rotational Equilibrium3h 39m
16. Angular Momentum3h 6m
17. Periodic Motion2h 9m
18. Waves & Sound3h 40m
19. Fluid Mechanics4h 27m
20. Heat and Temperature3h 7m
21. Kinetic Theory of Ideal Gases1h 50m
22. The First Law of Thermodynamics1h 26m
23. The Second Law of Thermodynamics3h 11m
24. Electric Force & Field; Gauss' Law3h 42m
25. Electric Potential1h 51m
26. Capacitors & Dielectrics2h 2m
27. Resistors & DC Circuits3h 8m
28. Magnetic Fields and Forces2h 23m
29. Sources of Magnetic Field2h 30m
30. Induction and Inductance3h 38m
31. Alternating Current2h 37m
32. Electromagnetic Waves2h 14m
33. Geometric Optics2h 57m
34. Wave Optics1h 15m
35. Special Relativity2h 10m

4. 2D Kinematics

Intro to Relative Velocity

Physics

4. 2D Kinematics

Intro to Relative Velocity

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4:18 minutes

Problem 35a

Textbook Question

A river flows due south with a speed of 2.0 m/s. You steer a motorboat across the river; your velocity relative to the water is 4.2 m/s due east. The river is 500 m wide. What is your velocity (magnitude and direction) relative to the earth?

Verified step by step guidance

First, identify the components of the boat's velocity relative to the earth. The boat has a velocity of 4.2 m/s due east relative to the water and the river flows south at 2.0 m/s. These two velocities are perpendicular to each other, forming a right triangle.

Use the Pythagorean theorem to find the magnitude of the resultant velocity. The magnitude $ v $ of the velocity relative to the earth can be calculated using $ v = \sqrt{v_{east}^2 + v_{south}^2} $, where $ v_{east} = 4.2 $ m/s and $ v_{south} = 2.0 $ m/s.

Calculate the direction of the resultant velocity using trigonometry. The direction $ \theta $ can be found using the tangent function: $ \tan(\theta) = \frac{v_{south}}{v_{east}} $. Solve for $ \theta $ to find the angle south of east.

Now, consider the width of the river, which is 500 m. Calculate the time it takes to cross the river using the eastward component of the velocity: $ t = \frac{d}{v_{east}} $, where $ d = 500 $ m.

Finally, use the time calculated to determine how far downstream the boat will be when it reaches the opposite bank. Multiply the southward component of the velocity by the time: $ d_{downstream} = v_{south} \times t $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Relative Velocity

Relative velocity is the velocity of an object as observed from a particular reference frame. In this problem, the boat's velocity relative to the earth is determined by combining its velocity relative to the water and the river's velocity. This involves vector addition, where the boat's eastward velocity and the river's southward velocity are combined to find the resultant velocity.

Vector Addition

Vector addition is a method used to combine two or more vectors to find a resultant vector. It involves adding the components of the vectors along each axis. In this scenario, the boat's eastward velocity and the river's southward velocity are perpendicular, requiring the use of the Pythagorean theorem to find the magnitude of the resultant velocity and trigonometry to determine its direction.

Trigonometry in Physics

Trigonometry is used in physics to resolve vectors into components and to find angles between vectors. In this problem, trigonometric functions such as sine, cosine, or tangent can be used to determine the direction of the resultant velocity vector relative to the earth. The angle can be calculated using the inverse tangent function, considering the ratio of the river's velocity to the boat's velocity.

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Related Practice

Textbook Question

The nose of an ultralight plane is pointed due south, and its airspeed indicator shows $35\text{ m/s}$ . The plane is in a $10\text{ m/s}$ wind blowing toward the southwest relative to the earth. In a vector-addition diagram, show the relationship of $\overrightarrow{v}_{P/E}$ (the velocity of the plane relative to the earth) to the two given vectors.

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Textbook Question

A 'moving sidewalk' in an airport terminal moves at 1.0 m/s and is 35.0 m long. If a woman steps on at one end and walks at 1.5 m/s relative to the moving sidewalk, how much time does it take her to reach the opposite end if she walks In the opposite direction?

A river flows due south with a speed of 2.0 m/s. You steer a motorboat across the river; your velocity relative to the water is 4.2 m/s due east. The river is 500 m wide. How much time is required to cross the river?

A canoe has a velocity of 0.40 m/s southeast relative to the earth. The canoe is on a river that is flowing 0.50 m/s east relative to the earth. Find the velocity (magnitude and direction) of the canoe relative to the river.

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Textbook Question

An airplane pilot wishes to fly due west. A wind of 80.0 km/h (about 50 mi/h) is blowing toward the south. If the airspeed of the plane (its speed in still air) is 320.0 km/h (about 200 mi/h), in which direction should the pilot head?

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Textbook Question

The nose of an ultralight plane is pointed due south, and its airspeed indicator shows $35 m slash s$ . The plane is in a $10 m slash s$ wind blowing toward the southwest relative to the earth. Let $x$ be east and $y$ be north, and find the components of ${\vec{v}}_{P / E}$ .

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Textbook Question

(II) In what direction should the pilot aim the plane in Problem 67 so that it will fly due south?

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