Chapter 1: Problem 62
A plane leaves the airport in Galisteo and flies 170 km at 68.0\(^{\circ}\) east of north; then it changes direction to fly 230 km at 36.0\(^{\circ}\) south of east, after which it makes an immediate emergency landing in a pasture. When the airport sends out a rescue crew, in which direction and how far should this crew fly to go directly to this plane?
Short Answer
Step by step solution
Resolve First Leg of the Journey
Resolve Second Leg of the Journey
Calculate Net Displacement Components
Find Total Displacement and Direction
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Displacement Components
Each movement of the plane can be seen as a vector, which has both magnitude (distance) and direction (angle). When a plane travels at an angle, like north-east or south-east, we don't directly know how far it goes in the north or east direction separately. Instead, we resolve the vector into its components.
- Northern Component: How far the plane travels northward.
- Eastern Component: How far the plane travels eastward.
- Southern Component: How far the plane travels southward, if reversing direction.
Applying the Pythagorean Theorem
In the context of vector addition for displacement, once we have the net components in each direction, we can use the Pythagorean theorem to find the total displacement. Here’s how:
- Net Eastern and Northern/Southern Components: Once we have resolved all vectors into their components, we'll have a net eastern and a net northern or southern component.
- Calculate Hypotenuse (Total Displacement): Apply the formula \( d = \sqrt{(x)^2 + (y)^2} \) where \( x \) and \( y \) are the net east and south/north components.
Role of Trigonometric Functions
- Sine (\(\sin\)): In a right triangle, it is the ratio of the length of the opposite side to the hypotenuse; useful for finding vertical components from angles.
- Cosine (\(\cos\)): This is the ratio of the length of the adjacent side to the hypotenuse; used for horizontal components from angles.
- Tangent (\(\tan\)): The ratio of the opposite side to the adjacent side; ideal for finding an angle when you know two sides.