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- Thread starter rashida564
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If the object is at constant speed, then all accelerations are perpendicular to the current velocity. A simple example would be a car moving at constant speed, with the driver adjusting steering inputs so that the car follows an elliptical path (imagine doing this on an elliptical shaped loop of road).

Trying to create an equation for this is complicated. It may require parametric equations defining x and y as functions of time.

Trying to create an equation for this is complicated. It may require parametric equations defining x and y as functions of time.

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$$\vec{x}=\begin{pmatrix} a \cos \phi \\ b \sin \phi \end{pmatrix}.$$

Then

$$\vec{v}=\begin{pmatrix} -a \sin \phi \\ b \cos \phi \end{pmatrix} \dot{\phi}.$$

The equation thus is

$$\vec{v}^2=\dot{\phi}^2 (a^2 \sin^2 \phi + b^2 \cos^2 \phi)=\text{const}.$$

or

$$\mathrm{d}_t \vec{v}^2 = 0 \; \Rightarrow \; \dot{\phi} \ddot{\phi} (a^2 \sin^2 \phi + b^2 \cos^2 \phi) + \dot{\phi}^3 (a^2-b^2) \sin \phi \cos \phi =0.$$

This is indeed very difficult to solve. Mathematica gives some complicated implicit solution with some elliptic functions ;-)).

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What is φ in your equations? I'm wondering if it would help to express x and y as functions of time. I did a web search and found a paper for ellipse with constant speed, but they're asking $16.00 to download it, and it's possible that the paper doesn't come up with an actual solution.This is indeed very difficult to solve.

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Here is the link (maybe you can find this elsewhere). I did a search for "ellipse parametric constant speed" to search for articles, and this seemed to be the only one that could have a solution. It's from a math magazine article, archived at this web site.Which paper is it? Maybe I can download it via my university account.

https://www.jstor.org/stable/10.4169/math.mag.86.1.003?seq=1

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