Difference between revisions of "009A Sample Midterm 1, Problem 5"
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!Foundations: | !Foundations: | ||
|- | |- | ||
| − | |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? | + | |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 22: | Line 22: | ||
!Step 1: | !Step 1: | ||
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| − | |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 style="vertical-align: -5px">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 | ||
| Line 54: | Line 54: | ||
\end{array}</math> | \end{array}</math> | ||
|- | |- | ||
| − | |Therefore, the velocity of the object at time <math>t=\frac{\pi}{8}</math> is | + | |Therefore, the velocity of the object at time <math>t=\frac{\pi}{8}</math> is |
|- | |- | ||
| <math>\begin{array}{rcl} | | <math>\begin{array}{rcl} | ||
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!Final Answer: | !Final Answer: | ||
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| − | | position is <math>\frac{1}{4} \text{ foot}.</math> | + | | position is <math>\frac{1}{4} \text{ foot}.</math> |
|- | |- | ||
| − | | velocity is <math>4 \text{ feet/second}.</math> | + | | velocity is <math>4 \text{ feet/second}.</math> |
|} | |} | ||
[[009A_Sample_Midterm_1|'''<u>Return to Sample Exam</u>''']] | [[009A_Sample_Midterm_1|'''<u>Return to Sample Exam</u>''']] | ||
Revision as of 15:52, 26 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 (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 4 \text{ feet/second}.} |