009A Sample Midterm 1, Problem 5

The displacement from equilibrium of an object in harmonic motion on the end of a spring is:

$y={\frac {1}{3}}\cos(12t)-{\frac {1}{4}}\sin(12t)$

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  $t={\frac {\pi }{8}}.$

Foundations:
What is the relationship between the position  $s(t)$  and the velocity  $v(t)$  of an object?
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle v(t)=s'(t)}

Solution:

Step 1:
To find the position of the object at  $t={\frac {\pi }{8}},$
we need to plug  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle t={\frac {\pi }{8}}}   into the equation  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y.}
Thus, we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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
${\begin{array}{rcl}\displaystyle {v(t)}&=&\displaystyle {y'}\\&&\\&=&\displaystyle {{\frac {-1}{3}}\sin(12t)(12)-{\frac {1}{4}}\cos(12t)(12)}\\&&\\&=&\displaystyle {-4\sin(12t)-3\cos(12t).}\end{array}}$
Therefore, the velocity of the object at time  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle t={\frac {\pi }{8}}}   is
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {1}{4}}{\text{ foot}}.}
velocity is  $4{\text{ feet/second}}.$

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