Textbook Question
The position of a squirrel running in a park is given by r = [(0.280 m/s)t + (0.0360 m/s2)t2]î + (0.0190 m/s3)t3ĵ. (a) What are υx(t) and υy(t), the x- and y-components of the velocity of the squirrel, as functions of time?
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Ch 03: Motion in Two or Three Dimensions
Chapter 3, Problem 3
An airplane pilot wishes to fly due west. A wind of 80.0 km/h (about 50 mi/h) is blowing toward the south. (a) 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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Identify the given values: wind speed is 80.0 km/h south, and the plane's airspeed is 320.0 km/h.
Set up a vector diagram: Represent the wind velocity vector pointing south and the plane's airspeed vector pointing in the unknown direction that the pilot should head.
Use vector addition to find the resultant vector, which should point due west. The resultant vector is the vector sum of the plane's airspeed vector and the wind velocity vector.
Apply the sine and cosine rules to solve for the angle at which the pilot should head. This involves calculating the angle between the resultant vector (due west) and the airspeed vector of the plane.
Adjust the plane's heading to compensate for the wind. The angle calculated will tell the pilot the direction to head relative to north (or south) to ensure the actual flight path is due west.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Addition
Vector addition is the process of combining two or more vectors to determine a resultant vector. In this scenario, the airplane's velocity and the wind's velocity are both vectors, with specific magnitudes and directions. The pilot must calculate the resultant vector to ensure the airplane travels due west, taking into account the southward wind.
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Guided course
07:30
Vector Addition By Components
Relative Velocity
Relative velocity refers to the velocity of an object as observed from a particular reference frame. In this case, the pilot's airspeed is relative to the air, while the wind's speed is relative to the ground. Understanding relative velocity is crucial for determining the correct heading to counteract the wind's effect and maintain the desired course.
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04:27Intro to Relative Motion (Relative Velocity)
Trigonometry in Navigation
Trigonometry is essential in navigation for resolving vectors into components. The pilot can use trigonometric functions to find the angle at which to head the airplane to counteract the wind. By applying sine and cosine functions, the pilot can determine the necessary heading to achieve a westward trajectory despite the southward wind.
Recommended video:
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3:26Trigonometry
Related Practice
Textbook Question
The coordinates of a bird flying in the xy-plane are given by x(t) = αt and y(t) = 3.0 m − βt2, where α = 2.4 m/s and β = 1.2 m/s2. (b) Calculate the velocity and acceleration vectors of the bird as functions of time.
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Textbook Question
The nose of an ultralight plane is pointed due south, and its airspeed indicator shows 35 m/s. The plane is in a 10–m/s wind blowing toward the southwest relative to the earth. (b) Let x be east and y be north, and find the components of υ→ P/E.
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Textbook Question
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
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. (a) What is your velocity (magnitude and direction) relative to the earth?
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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. (b) How much time is required to cross the river?
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