Difference between revisions of "009A Sample Midterm 1, Problem 5"
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::<span class="exam"><math>y=\frac{1}{3}\cos(12t)-\frac{1}{4}\sin(12t)</math> | ::<span class="exam"><math>y=\frac{1}{3}\cos(12t)-\frac{1}{4}\sin(12t)</math> | ||
| − | <span class="exam">where <math>y</math> is measured in feet and <math>t</math> is the time in seconds. | + | <span class="exam">where <math style="vertical-align: -4px">y</math> is measured in feet and <math style="vertical-align: 0px">t</math> is the time in seconds. |
| − | <span class="exam">Determine the position and velocity of the object when <math>t=\frac{\pi}{8}.</math> | + | <span class="exam">Determine the position and velocity of the object when <math style="vertical-align: -14px">t=\frac{\pi}{8}.</math> |
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!Foundations: | !Foundations: | ||
|- | |- | ||
| − | |What is the relationship between position <math>s(t)</math> and velocity <math>v(t)</math> of an object? | + | |What is the relationship between position <math style="vertical-align: -5px">s(t)</math> and velocity <math style="vertical-align: -5px">v(t)</math> of an object? |
|- | |- | ||
| <math>v(t)=s'(t)</math> | | <math>v(t)=s'(t)</math> | ||
| Line 24: | Line 24: | ||
|To find the position of the object at <math>t=\frac{\pi}{8},</math> | |To find the position of the object at <math>t=\frac{\pi}{8},</math> | ||
|- | |- | ||
| − | |we need to plug <math>t=\frac{\pi}{8}</math> into the equation <math>y.</math> | + | |we need to plug <math>t=\frac{\pi}{8}</math> into the equation <math style="vertical-align: -5px">y.</math> |
|- | |- | ||
|Thus, we have | |Thus, we have | ||
Revision as of 15:59, 18 February 2017
The displacement from equilibrium of an object in harmonic motion on the end of a spring is:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=\frac{1}{3}\cos(12t)-\frac{1}{4}\sin(12t)}
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} is measured in feet and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} is the time in seconds.
Determine the position and velocity of the object when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=\frac{\pi}{8}.}
| Foundations: |
|---|
| What is the relationship between position Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s(t)} and velocity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(t)} of an object? |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(t)=s'(t)} |
Solution:
| Step 1: |
|---|
| To find the position of the object at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=\frac{\pi}{8},} |
| we need to plug Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=\frac{\pi}{8}} into the equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y.} |
| Thus, we have |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{y\bigg(\frac{\pi}{8}\bigg)} & = & \displaystyle{\frac{1}{3}\cos\bigg(\frac{12\pi}{8}\bigg)-\frac{1}{4}\sin\bigg(\frac{12\pi}{8}\bigg)}\\ &&\\ & = & \displaystyle{\frac{1}{3}\cos\bigg(\frac{3\pi}{2}\bigg)-\frac{1}{4}\sin\bigg(\frac{3\pi}{2}\bigg)}\\ &&\\ & = & \displaystyle{0-\frac{1}{4}(-1)}\\ &&\\ &= & \displaystyle{\frac{1}{4} \text{ foot}.} \end{array}} |
| Step 2: |
|---|
| Now, to find the velocity function, we need to take the derivative of the position function. |
| Thus, we have |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{v(t)} & = & \displaystyle{y'}\\ &&\\ & = & \displaystyle{\frac{-1}{3}\sin(12t)(12)-\frac{1}{4}\cos(12t)(12)}\\ &&\\ & = & \displaystyle{-4\sin(12t)-3\cos(12t).} \end{array}} |
| Therefore, the velocity of the object at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=\frac{\pi}{8}} is |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{v\bigg(\frac{\pi}{8}\bigg)} & = & \displaystyle{-4\sin\bigg(\frac{3\pi}{2}\bigg)-3\cos\bigg(\frac{3\pi}{2}\bigg)}\\ &&\\ & = & \displaystyle{-4(-1)+0}\\ &&\\ & = & \displaystyle{4 \text{ feet/second}.} \end{array}} |
| Final Answer: |
|---|
| position is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{4} \text{ foot}.} |
| velocity is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 4{\text{ feet/second}}.} |