help with wagon wheel tractrix

[puffers]

I have spent a long time on problem #2, but I cannot figure it out. The answer key says the answer is 1/sqrt(1-y^2), but I have no idea how they got there. I originally thought that Jamie would just have a speed of sqrt(1-y^2), the same as the dx/dt for the wagon because they are attached. I also provided problem #1 for context. Thanks in advance!

#1:
Walking along the x-axis, Jamie uses a rope of unit length to drag a wagon W that is initially at (0, 1). Thus W = (x,y) rolls along a tractrix. Suppose that Jamie walks in such a way that the speed of W is 1 unit per second. It follows that the velocity component dy/dt of the wagon is exactly −y. Explain why this is so. You should also be able to express the component dx/dt in terms of y. Show that y = e^−t.
*I already solved this problem, but the info it provides is important for #674*

#2:
(Continuation) Given that W = (x, y), explain why J = (x + 1/sqrt(1-y^2) , 0). Use this
equation to calculate a formula for Jamie’s speed, as a function of y.

 

[Dr.Peterson]

I have spent a long time on problem #2, but I cannot figure it out. The answer key says the answer is 1/sqrt(1-y^2), but I have no idea how they got there. I originally thought that Jamie would just have a speed of sqrt(1-y^2), the same as the dx/dt for the wagon because they are attached. I also provided problem #1 for context. Thanks in advance!

#1:
Walking along the x-axis, Jamie uses a rope of unit length to drag a wagon W that is initially at (0, 1). Thus W = (x,y) rolls along a tractrix. Suppose that Jamie walks in such a way that the speed of W is 1 unit per second. It follows that the velocity component dy/dt of the wagon is exactly −y. Explain why this is so. You should also be able to express the component dx/dt in terms of y. Show that y = e^−t.
*I already solved this problem, but the info it provides is important for #674*

#2:
(Continuation) Given that W = (x, y), explain why J = (x + 1/sqrt(1-y^2) , 0). Use this
equation to calculate a formula for Jamie’s speed, as a function of y.

Could you show us your work for both parts? That will help us correct whatever you did to get the wrong answer.

But the answer to your specific question is that the x-component of the velocities of W and J are different because the segment joining them rotates as they move. They are not separated by a constant horizontal distance.

 

[Dr.Peterson]

#2:
(Continuation) Given that W = (x, y), explain why J = (x + 1/sqrt(1-y^2) , 0). Use this
equation to calculate a formula for Jamie’s speed, as a function of y.

I checked to see if you copied the problem correctly, and found it on another site:

​

Is this really what your version says? Is that why you got the wrong answer?

 

[mario99]

I have spent a long time on problem #2, but I cannot figure it out. The answer key says the answer is 1/sqrt(1-y^2), but I have no idea how they got there. I originally thought that Jamie would just have a speed of sqrt(1-y^2), the same as the dx/dt for the wagon because they are attached. I also provided problem #1 for context. Thanks in advance!

Professor Dave has given you the correct form of [imath]J[/imath]:

[imath]\displaystyle J = \left(x + \sqrt{1 - y^2}, 0\right)[/imath]

In problem [imath]666[/imath], you have already shown that [imath]y = e^{-t}[/imath] and you should already know that:

[imath]x(t) = t + \ln\left(1 + \sqrt{1 - e^{-2t}}\right) - \sqrt{1 - e^{-2t}} \ \ \ \ [/imath] (From the solution of the Tractrix Differential Equation)

This gives us:

[imath]\displaystyle J(t) = \left(t + \ln\left(1 + \sqrt{1 - e^{-2t}}\right) - \sqrt{1 - e^{-2t}} + \sqrt{1 - e^{-2t}}, 0\right) = \left(t + \ln\left(1 + \sqrt{1 - e^{-2t}}\right), 0\right)[/imath]

The speed of Jamie is:

[imath]\displaystyle \frac{J(t)}{dt} = \frac{1}{\sqrt{1 - e^{-2t}}} = \frac{1}{\sqrt{1 - y^{2}}}[/imath]

 

[puffers]

I checked to see if you copied the problem correctly, and found it on another site:

View attachment 38389​

Is this really what your version says? Is that why you got the wrong answer?

Oops! I totally copied it down wrong when I was typing, but I still used the same J as you when I was doing my work. Here is what I did, honestly I think I'm just confusing myself at this point. Thanks for your help!

 

[mario99]

Wagon wheel tractrix

I have spent a long time on problem #2, but I cannot figure it out. The answer key says the answer is 1/sqrt(1-y^2), but I have no idea how they got there. I originally thought that Jamie would just have a speed of sqrt(1-y^2), the same as the dx/dt for the wagon because they are attached. I...

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