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Our teacher told us today:

Suppose a satellite is moving around the earth in a circular orbit of radius $r$. Suddenly an external agent stops its motion so that its kinetic energy becomes zero but its gravitational potential energy remains the same. Assume there is no air friction or drag. The satellite starts to fall towards the earth's surface and it will follow an elliptical orbit in its motion towards the earth.

Please tell me if this is correct because according to my intuition it should follow a straight-line trajectory towards the earth.

Please help me with this. I am not very good at English so pardon me for any grammatical mistakes in my question.

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    $\begingroup$ Note that an ellipse whose "minor axis" is zero is just a line segment. A possibly-related answer. My inner mathematician wants to get all torn up about whether an ellipse in the limit where eccentricity $e\to 1$ is a line segment or a parabola, but my outer physicist isn't worried about the difference. $\endgroup$
    – rob ♦
    Commented Aug 15, 2020 at 22:28

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Short answer: you are both right.

Suppose that the satellite did not stop completely, but still have some very-very-very small speed. The orbit of this satellite will be a very stretched ellipse. It will not be much different from the straight line, but still an ellipse.

So, it is possible (and convenient!) to consider the straight line towards Earth's center as a degenerate ellipse.

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  • $\begingroup$ thankyou so much $\endgroup$
    – gurdeep singh
    Commented Aug 15, 2020 at 20:37
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If the satelite is slightly accelerated or decelerated, its trajectory will change to an elipse.

But if it is stopped, that is: no angular velocity with the center of the earth, it will fall radially.

If it starts to fall over one of the poles, its trajectory will be vertical to an observer on earth. Otherwise, it will follow (for a observer on earth) a curved shape (I see no reason to be an elipse) depending on the latitude and the initial height.

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The satellite will stop but the rotation of the Earth no. The satellite will fall vertically but the trajectory of the satellite will be an ellipse relative to the vertical point where the satellite was stopped because of the rotation of the Earth.

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  • $\begingroup$ okk ...got your point...thankyou $\endgroup$
    – gurdeep singh
    Commented Aug 15, 2020 at 20:49
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    $\begingroup$ I would not agree with you here :) Rotation of the earth around it's axis does not affect the satellite. Imagine an astronaut who observes the satellite and the Earth from some distance. He would see that the satellite moves along some ellipse trajectory, the Earth is rotating. If something would stop the Earth, or make it rotate faster - satellite's trajectory would not change at all. Well, if you stay on Earth and observe the satellite in this rotating frame of reference - yes, you visible satellite trajectory will change. But in the rotating frame of reference it can be not even an ellipse. $\endgroup$
    – lesnik
    Commented Aug 15, 2020 at 21:51
  • $\begingroup$ As Mr Saspinski says, when the sattelite stops it will fall on that vertical RADIALLY (I said eliptically) because the earth is rotating, or it will have a curved (not necessarily eliptical) trajectory if the satellite did not stop completely. $\endgroup$
    – user36636
    Commented Aug 17, 2020 at 6:55

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