0

Is it possible to obtain a time-dependent coordinate of an object moving at a constant velocity $v$ over an ellipse $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1?$$

Aig
  • 5,725
hhyy hhyy
  • 1
  • Welcome to MathSE! Please see how to ask a good question. It’s better to show your attempt in the post. What progress did you achieve? Where were you “stuck”? – Aig Apr 14 '24 at 03:45
  • 2
    Do you mean constant speed? Constant velocity implies straight line motion with uniform speed. – Ng Chung Tak Apr 14 '24 at 03:49
  • 2
    You need to parametrize by arc length but unfortunately, there is no closed form solution of parametrization of ellipse by arc length. Please refer to another post here. – Ng Chung Tak Apr 14 '24 at 03:53
  • Thank you for your kind words. The original purpose was to install a sound detector at one focal point of the ellipse and calculate the Doppler effect over time when the sound source moves at a constant speed along the elliptical orbit. I tried to differentiate the distance between the focal point and a point in the ellipse with respect to time for this. So I wanted to express the position over time. So I wanted to express the integral expression like time times velocity is the length of arc, but it was too complicated. I posted this because I was wondering if you had any other ideas – hhyy hhyy Apr 14 '24 at 04:29
  • 1
    This information is important and should be in the post, not in the comment. – Aig Apr 14 '24 at 05:10
  • Do you mean something like x and y can be represented as a parameter using some form of t? – Gwen Apr 14 '24 at 05:22
  • @Gwen - that would be difficult considering the requirement that speed is constant, in other words, the arc length for each time increment needs to be the same length... if not for that constraint, yes, quite easy and there are a few options using different trigonometric functions. – Red Five Apr 14 '24 at 06:29
  • @RedFive is there some way where we can use Kepler's laws for planets? (It may seem I am joking :") but I'm serious) – Gwen Apr 14 '24 at 06:41
  • Yes, interesting... maybe. The only issue is an ellipse has two foci and Kepler's laws refer to speed relative only to one of the foci so not likely to work unless you can see a way around this. – Red Five Apr 14 '24 at 06:59
  • The ellipse is parametrized as $p(t) = (x(t), y(t)) = (a \cos \theta , b \sin \theta)$, the speed is given by $ s = \dot{\theta} \sqrt{ a^2 \sin^2 \theta + b^2 \cos^2 \theta } $. Integrating this from $t = 0$ gives $ \theta(t) $. –  Apr 14 '24 at 08:11

0 Answers0