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I know that for an object to maintain circular motion, it needs to have a specific linear speed ( if you we keep Centripetal Force and the Radius constant). If the object travels too fast, it will fly out of orbit and if it travels too slow it will fall in towards the direction of acceleration. But my question is, mathematically and theoretically, what really happens when an object travels too fast or slow that it falls in and out of its circular motion or orbit? Does it have anything to do with the change in momentum?

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It depends on the force between the object and the central point. Suppose our object is moving with some speed $v$ at a distance $r$ from the centre:

Circular motion

There is some force $F$ between the object and the central point. This force could be gravity if we are talking about a planet going round a star, or it could be tension in a string if we are whirling a stone around our head. All we can say is that for a circular orbit this force must be balanced out by the centrifugal force $mv^2/r$ so we get:

$$ F(r) = \frac{mv^2}{r} $$

So for example if we are talking about a planet orbiting a star of mass $M$ then the force is the gravitational force:

$$ F(r) = \frac{GMm}{r^2} $$

so for a circular orbit we get:

$$ \frac{GMm}{r^2} = \frac{mv^2}{r} $$

giving us the usual equation for the velocity in a circular orbit:

$$ v = \sqrt{\frac{GM}{r}} $$

But the point of your question is what happens if the object is moving slower or faster. And the answer is that in this case the force $F(r)$ will be greater or less than the centrifugal force and the object will move inwards or outwards respectively i.e. it will move along some path that isn't circular.

Exactly what happens depends on how our force $F(r)$ varies with distance. If we go back to our example of a planet orbiting a star then the trajectory will be an ellipse. Suppose the velocity is less than the circular velocity, then the gravitational force will be lower than the centrifugal force and the planet will fall inwards. The orbit will look like this:

Elliptical orbit

Since the velocity $v_a$ is less than the circular velocity the planet falls inwards. However as it falls inwards it speeds up, and at the closest point its velocity $v_p$ is now greater than the circular orbit velocity so it starts moving outwards again and eventually returns to the point it started. Overall the orbit is an ellipse. That's why gravitational orbits are always ellipses.

But we only get ellipses when the force is proportional to $1/r^2$ like gravity. There could be different forces e.g. suppose you're whirling a stone around your head but on the end of a spring that can stretch to allow the stone to move inwards and outwards. This would give different trajectories. Note also that for gravity the orbits are always stable, which is just as well really since it would be awkward if the Earth were to suddenly fall into the Sun. But for different types of force the orbits aren't necessarily stable.

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  • $\begingroup$ The OP asked about momentum. Perhaps you could include what effect that has? $\endgroup$ – sammy gerbil Jan 2 '18 at 21:00

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