A box (of negligible size) slides down a smooth parabolic ramp. A stretchy cord (which acts like a spring) is attached to the box. The cord has a spring constant of k = 100 N/m and a natural length of Ln = 1.8 m. Initially, the box starts at rest at point A. The initial coordinates of the box are given by (1.5, 2.25) m. At point B, the position is given by the coordinates (2, 1) m. The box has a mass of 7 kg. Based off this information, answer the following questions:i) What is the speed of the box at location B?ii) What is the velocity of the box at location B?iii) What is the normal force from the ramp acting on the box at location B? ### Parabolic Graph IllustrationThe provided diagram illustrates a parabola represented by the equation \( y = (x-3)^2 \) meters. This has a typical "U" shape characteristic of parabolas.#### Components of the Diagram:1. **Axes:** - The horizontal axis is labeled as \( x \). - The vertical axis is labeled as \( y \). - The origin of the coordinate system is denoted by the point \( O \).2. **Parabolic Curve:** - The curve represents the parabola defined by the equation \( y = (x-3)^2 \). This equation suggests that the vertex of the parabola is at the point \( (3, 0) \) on the coordinate plane because the expression \( (x-3) \) gets squared.3. **Points and Tangents:** - Point \( A \) and Point \( B \) are marked on the parabola. - At Point \( B \), a normal (perpendicular) line to the parabola is drawn, marked with a right angle symbol to highlight that it’s perpendicular to the tangential line at that point. - A similar perpendicular line is suggested at Point \( A \) but represented with a dashed line, indicating the position of the tangent and the normal at Point \( A \).4. **Axis Labels and Arrowheads:** - The axes extend with arrows indicating the positive directions. - The coordinate axes intersect at the origin, \( O \).This image serves as a graphical representation of basic concepts in coordinate geometry and calculus, including the identification of parabolic shapes, their equations, and the geometric interpretation of tangents and normals to curves.

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Answered: A box (of negligible size) slides down…

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 box (of negligible size) slides down a smooth parabolic ramp. A stretchy cord (which acts like a spring) is attached to the box. The cord has a spring constant of k = 100 N/m and a natural length of Ln = 1.8 m. Initially, the box starts at rest at point A. The initial coordinates of the box are given by (1.5, 2.25) m. At point B, the position is given by the coordinates (2, 1) m. The box has a mass of 7 kg. Based off this information, answer the following questions:

i) What is the speed of the box at location B?

ii) What is the velocity of the box at location B?

iii) What is the normal force from the ramp acting on the box at location B?

### Parabolic Graph Illustration

The provided diagram illustrates a parabola represented by the equation \( y = (x-3)^2 \) meters. This has a typical "U" shape characteristic of parabolas.

#### Components of the Diagram:
1. **Axes:**
- The horizontal axis is labeled as \( x \).
- The vertical axis is labeled as \( y \).
- The origin of the coordinate system is denoted by the point \( O \).

2. **Parabolic Curve:**
- The curve represents the parabola defined by the equation \( y = (x-3)^2 \). This equation suggests that the vertex of the parabola is at the point \( (3, 0) \) on the coordinate plane because the expression \( (x-3) \) gets squared.

3. **Points and Tangents:**
- Point \( A \) and Point \( B \) are marked on the parabola.
- At Point \( B \), a normal (perpendicular) line to the parabola is drawn, marked with a right angle symbol to highlight that it’s perpendicular to the tangential line at that point.
- A similar perpendicular line is suggested at Point \( A \) but represented with a dashed line, indicating the position of the tangent and the normal at Point \( A \).

4. **Axis Labels and Arrowheads:**
- The axes extend with arrows indicating the positive directions.
- The coordinate axes intersect at the origin, \( O \).

This image serves as a graphical representation of basic concepts in coordinate geometry and calculus, including the identification of parabolic shapes, their equations, and the geometric interpretation of tangents and normals to curves.

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