Given the following displacement vector (and thus, also, acceleration vector):(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\begin{array}{l}

\overrightarrow r (t) = \left( {\begin{array}{*{20}c}

{1 + 3\cos (\frac{{\pi t}}{2})} \\

{2 + 4\sin (\frac{{\pi t}}{2})} \\

\end{array}} \right) \\

\overrightarrow a (t) = \left( {\begin{array}{*{20}c}

{\frac{{ - 3\pi ^2 }}{4}\cos (\frac{{\pi t}}{2})} \\

{ - \pi ^2 \sin (\frac{{\pi t}}{2})} \\

\end{array}} \right) \\

\end{array}

[/tex]

I need to show that the body is undergoing eliptical motion. This is how i proceeded:

[tex]

\begin{array}{l}

\overrightarrow r (t) = \left( {\begin{array}{*{20}c}

{1 + 3\cos (\frac{{\pi t}}{2})} \\

{2 + 4\sin (\frac{{\pi t}}{2})} \\

\end{array}} \right) = \left( {\begin{array}{*{20}c}

{3\cos (\frac{{\pi t}}{2})} \\

{4\sin (\frac{{\pi t}}{2})} \\

\end{array}} \right) + \left( {\begin{array}{*{20}c}

1 \\

2 \\

\end{array}} \right) \\

\therefore\left( {\begin{array}{*{20}c}

{3\cos (\frac{{\pi t}}{2})} \\

{4\sin (\frac{{\pi t}}{2})} \\

\end{array}} \right) = \overrightarrow r (t) - \left( {\begin{array}{*{20}c}

1 \\

2 \\

\end{array}} \right) \\

\\

\overrightarrow a (t) = \left( {\begin{array}{*{20}c}

{\frac{{ - 3\pi ^2 }}{4}\cos (\frac{{\pi t}}{2})} \\

{ - \pi ^2 \sin (\frac{{\pi t}}{2})} \\

\end{array}} \right) = \frac{{ - \pi ^2 }}{4}\left( {\begin{array}{*{20}c}

{3\cos (\frac{{\pi t}}{2})} \\

{4\sin (\frac{{\pi t}}{2})} \\

\end{array}} \right) = \frac{{ - \pi ^2 }}{4}\left[ {\overrightarrow r (t) - \left( {\begin{array}{*{20}c}

1 \\

2 \\

\end{array}} \right)} \right] \\

\end{array}

[/tex]

Is that how i should do it? By showing that the acceleration is proportional to the displacement vector?

Thanks in advance,

Dan.

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# Homework Help: Elliptical Motion

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