243.13.01.19 For which path is the particle's speed constant

In summary, the following equations each describe the motion of a particle. For which path is the particle's speed constant?
  • #1
karush
Gold Member
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$\tiny{243.13.01.19}$
$\textsf{The following equations each describe the motion of a particle.}$
$\textsf{ For which path is the particle's speed constant?}$
\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!:confused:
 
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  • #2
Here, you just want to find those choices where the magnitude of the velocity vector is constant. :)
 
  • #3
Isn't the magnitude of the velocity vector $\sqrt{2}$ which is a constant, so we don't need the the dot product?
 
  • #4
karush said:
Isn't the magnitude of the velocity vector $\sqrt{2}$ which is a constant, so we don't need the the dot product?

Yes, we don't need the dot product here. Your choice is correct, but I can see at least one other choice for which the speed is constant...:)
 
  • #5
I would guess $R_2$
the absense of powers > 1
 
  • #6
karush said:
I would guess $R_2$
the absense of powers > 1

All you're looking for is a velocity vector whose magnitude is constant, and the second choice isn't one of those. :)
 
  • #7
Not sure!
 
  • #8
I was mistaken...I "eyeballed" the 4th choice and saw the argument for the sine and cosine functions was the same, and mentally declared the velocity constant when in fact it varies as the magnitude of the parameter $t$ via the chain rule. Sorry for the hasty mistake. (Bandit)
 
  • #9
no prob
happens
 
Last edited:

Related to 243.13.01.19 For which path is the particle's speed constant

1. What does the numerical value in "243.13.01.19" represent?

The numerical value in "243.13.01.19" represents the specific date and time when the particle's speed was measured.

2. How is the particle's speed measured?

The particle's speed is measured using a variety of methods, such as using specialized equipment like speedometers or tracking its movement over a known distance and time.

3. What factors can affect the accuracy of the particle's speed measurement?

The accuracy of the particle's speed measurement can be affected by factors such as environmental conditions, errors in measurement equipment, and external forces acting on the particle.

4. Why is it important to know if the particle's speed is constant along a certain path?

Knowing if the particle's speed is constant along a certain path can help in understanding its motion and predicting its future behavior. It can also provide insights into the forces acting on the particle and the nature of its surroundings.

5. Can the particle's speed change along the path even if it is stated as constant?

Yes, the particle's speed can change along the path even if it is stated as constant. This could be due to external forces acting on the particle or changes in the environment that affect its motion.

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