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A small block slides down a frictionless track whose shape is described by y = (x^2) /d for x<0 and by y = -(x^2)/d for x>0. The value of d is 4.74 m, and x and y are measured in meters as usual. Now suppose the blocks starts on the track at x = 2.39 m. The block is given a push to the left and begins to slide up the track, eventually reaching its maximum height at x = 0, at which point it turns around and begins sliding down. What was its initial speed in this case? 6.74 m/s 4.86 m/s 3.44 m/s 4.98 m/s

A small block slides down a frictionless track whose shape is described by y = (x^2) /d for x<0 and by y = -(x^2)/d for x>0. The value of d is 4.74 m, and x and y are measured in meters as usual. Now suppose the blocks starts on the track at x = 2.39 m. The block is given a push to the left and begins to slide up the track, eventually reaching its maximum height at x = 0, at which point it turns around and begins sliding down. What was its initial speed in this case? 6.74 m/s 4.86 m/s 3.44 m/s 4.98 m/s

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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A small block slides down a frictionless track whose shape is described by y = (x^2) /d for x<0 and by y = -(x^2)/d for x>0. The value of d is 4.74 m, and x and y are measured in meters as usual. 

Now suppose the blocks starts on the track at x = 2.39 m. The block is given a push to the left and begins to slide up the track, eventually reaching its maximum height at x = 0, at which point it turns around and begins sliding down. What was its initial speed in this case?
   
6.74 m/s
   
4.86 m/s
   
3.44 m/s
   
4.98 m/s
### Understanding the Hyperbola In this lesson, we will explore the hyperbola, a fundamental concept in algebra and analytic geometry. A hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. #### Explanation of the Graph The image depicted above displays a simple hyperbola graph. Here's a detailed breakdown: 1. **Axes:** - The horizontal line is the \( x \)-axis. - The vertical line is the \( y \)-axis. - The point where these axes intersect is the origin \((0, 0)\). 2. **Hyperbola Curve:** - The curve in the graph is a representation of a hyperbola. - A hyperbola consists of two separate curves called "branches" which open in opposite directions. In this graph, one branch of the hyperbola opens towards the lower right quadrant, while the other opens towards the upper left quadrant. 3. **Behavior of the Curve:** - As we move along the curve further from the origin, the curve gets closer to the axes, but never touches them. This is a characteristic property of hyperbolas, where the axes are asymptotes for the branches. - The precise shape and orientation of the hyperbola depend on its defining equation. For example, the equation \(xy = c\) (with \( c \) being a constant) defines a rectangular hyperbola whose asymptotes are the coordinate axes. #### Key Properties of Hyperbolas: - **Asymptotes:** The axes in the graph function as asymptotes, approximating the curve at infinity but never intersecting. - **Vertices and Foci:** In a more detailed graph, hyperbolas have vertices and foci which play crucial roles in their geometric properties. #### Applications: Hyperbolas frequently occur in physics, engineering, and natural phenomena. Examples include the paths of comets around the sun in orbital mechanics and certain types of optical lenses. ### Conclusion This graph is a visual representation to aid in understanding the basic structure and properties of a hyperbola. In subsequent lessons, we will delve deeper into the algebraic equations and real-world applications of hyperbolas.
Transcribed Image Text:### Understanding the Hyperbola In this lesson, we will explore the hyperbola, a fundamental concept in algebra and analytic geometry. A hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. #### Explanation of the Graph The image depicted above displays a simple hyperbola graph. Here's a detailed breakdown: 1. **Axes:** - The horizontal line is the \( x \)-axis. - The vertical line is the \( y \)-axis. - The point where these axes intersect is the origin \((0, 0)\). 2. **Hyperbola Curve:** - The curve in the graph is a representation of a hyperbola. - A hyperbola consists of two separate curves called "branches" which open in opposite directions. In this graph, one branch of the hyperbola opens towards the lower right quadrant, while the other opens towards the upper left quadrant. 3. **Behavior of the Curve:** - As we move along the curve further from the origin, the curve gets closer to the axes, but never touches them. This is a characteristic property of hyperbolas, where the axes are asymptotes for the branches. - The precise shape and orientation of the hyperbola depend on its defining equation. For example, the equation \(xy = c\) (with \( c \) being a constant) defines a rectangular hyperbola whose asymptotes are the coordinate axes. #### Key Properties of Hyperbolas: - **Asymptotes:** The axes in the graph function as asymptotes, approximating the curve at infinity but never intersecting. - **Vertices and Foci:** In a more detailed graph, hyperbolas have vertices and foci which play crucial roles in their geometric properties. #### Applications: Hyperbolas frequently occur in physics, engineering, and natural phenomena. Examples include the paths of comets around the sun in orbital mechanics and certain types of optical lenses. ### Conclusion This graph is a visual representation to aid in understanding the basic structure and properties of a hyperbola. In subsequent lessons, we will delve deeper into the algebraic equations and real-world applications of hyperbolas.
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