- #1

karush

Gold Member

MHB

- 3,269

- 5

$\textsf{The following equations each describe the motion of a particle.}$

$\textsf{ For which path is the particle's speed constant?}$

\begin{align*} \displaystyle

R_1(t)&= t^7\textbf{i}+t^4\textbf{j}\\

R_2(t)&= \cos(3t)\textbf{i}+\sin(8t)\textbf{j}\\

R_3(t)&= t\textbf{i}+t\textbf{j}\\

R_4(t)&= \cos(3t^2)\textbf{i}+\sin(3t^2)\textbf{j}\\

%\textit{speed constant on}&=\color{red}{Path(3)}

\end{align*}

$\textit{By observation I would quess $ \, R_3(t)= t\textbf{i}+t\textbf{j}$ so then: }$

\begin{align*} \displaystyle

R_3(t)&= t\textbf{i}+t\textbf{j}\\

R_3^\prime (t)=V_3(t)&=\textbf{i}+\textbf{j}

\end{align*}

$\textit{so then dot product}$

\begin{align*}\displaystyle

\theta&=\cos^{-1}\left[\frac{u\cdot v}{|u||v|} \right] \\

&=\cos^{-1}\left[\frac{(t\textbf{i}+t\textbf{j})\cdot(\textbf{i}+\textbf{j})}

{|t\textbf{i}+t\textbf{j}||\textbf{i}+\textbf{j}|} \right]\\

\end{align*}

kinda maybe!