Difference between revisions of "009A Sample Midterm 1, Problem 5"

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|Review relationship between position and velocity
 
|Review relationship between position and velocity
 
|}
 
|}
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'''Solution:'''
 
'''Solution:'''
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!Step 1:    
 
!Step 1:    
 
|-
 
|-
−
|
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|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>
 +
|-
 +
|Thus, we have
 
|-
 
|-
−
|
+
|&nbsp; &nbsp; &nbsp; &nbsp; <math>\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)}\\
 +
&&\\
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& = & \displaystyle{\frac{1}{3}\cos\bigg(\frac{3\pi}{2}\bigg)-\frac{1}{4}\sin\bigg(\frac{3\pi}{2}\bigg)}\\
 +
&&\\
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& = & \displaystyle{0-\frac{1}{4}(-1)}\\
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&&\\
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&= & \displaystyle{\frac{1}{4} \text{ foot}.}
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\end{array}</math>
 
|}
 
|}
  
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
−
|
+
|Now, to find the velocity function, we need to take the derivative of the position function.
 +
|-
 +
|Thus, we have
 +
|-
 +
|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
 +
\displaystyle{v(t)} & = & \displaystyle{y'}\\
 +
&&\\
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& = & \displaystyle{\frac{-1}{3}\sin(12t)(12)-\frac{1}{4}\cos(12t)(12)}\\
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&&\\
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& = & \displaystyle{-4\sin(12t)-3\cos(12t).}
 +
\end{array}</math>
 +
|-
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|Therefore, the velocity of the object at time <math>t=\frac{\pi}{8}</math> is
 
|-
 
|-
−
|
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\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)}\\
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&&\\
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& = & \displaystyle{-4(-1)+0}\\
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&&\\
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& = & \displaystyle{4 \text{ feet/second}.}
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\end{array}</math>
 
|}
 
|}
  
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
−
|
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|&nbsp; &nbsp; &nbsp; &nbsp; position is <math>\frac{1}{4} \text{ foot}.</math>
 
|-
 
|-
−
|
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|&nbsp; &nbsp; &nbsp; &nbsp; 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 13:00, 16 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:  
Review relationship between position and velocity


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}.}

Return to Sample Exam

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