Doubling speed in circular orbit to attain elliptical orbit

In summary, the satellite's orbit is changed to a hyperbolic orbit with a semi-major axis of -R/2 after its speed is doubled for a short period of time. This is because the satellite's velocity after the propulsion is twice the escape velocity at that radius, resulting in an unbounded orbit. The given options for the semi-major axis in the multiple choice question are incorrect.
  • #1
ln(
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Homework Statement


A satellite is in a circular orbit (radius R) around a planet of mass M. To change the satellite's orbit the engines fire and its speed is suddenly doubled. The engines fire for a very short time. Determine the length of the semi-major axis of the new orbit.

Homework Equations


##v = \sqrt{GM(2/r - 1/a)}##
##\frac{-GMm}{2a} = \frac{1}{2}mv^2 - \frac{GMm}{r}##

The Attempt at a Solution


The velocity of the satellite prior to any propulsion is ##v = \sqrt{GM(2/r - 1/a)} = \sqrt{GM/R}## because ##a = r## for circular orbits.
Right after propulsion, we have ##2v = \sqrt{GM(2/r - 1/a)} = \sqrt{GM(2/R - 1/a)}##. If I square this, I get ##4v^2 = GM(2/R - 1/a) = 4GM/R## after substitution. Then, ##2/R - 1/a = 4/R## and ##a = -R/2##, but that isn't possible because ##a > 0## for elliptical orbits.

If I use energy, I end up with the same thing. Where have I gone wrong?
 
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  • #2
Perhaps you haven't done anything wrong...

Consider that escape speed is just ##\sqrt{2}## times the circular orbit speed at a given radius. Here you're doubling the speed. So what type of orbit should you end up with?
 
  • #3
gneill said:
Perhaps you haven't done anything wrong...

Consider that escape speed is just ##\sqrt{2}## times the circular orbit speed at a given radius. Here you're doubling the speed. So what type of orbit should you end up with?
An unbounded one?

EDIT: more specifically a hyperbolic one, where a < 0 is perfectly fine. That's interesting since this question actually is multiple choice in my notes, with answers choices a) R b) 1.5R c) 2R d) 2.5R or e) 4R. The key must be wrong then.
 
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  • #4
ln( said:
An unbounded one?
Yes indeed!
EDIT: more specifically a hyperbolic one, where a < 0 is perfectly fine. That's interesting since this question actually is multiple choice in my notes, with answers choices a) R b) 1.5R c) 2R d) 2.5R or e) 4R. The key must be wrong then.
Yup. The probably changed the question and forgot to update the answers.
 

Related to Doubling speed in circular orbit to attain elliptical orbit

1. How does doubling speed in circular orbit affect the shape of the orbit?

Doubling speed in circular orbit will cause the orbit to transition from a perfect circle to an elliptical shape. The orbit will become more elongated and the distance between the object and the center of the orbit will vary.

2. What is the significance of doubling speed in circular orbit to attain an elliptical orbit?

Doubling the speed in circular orbit is one way to achieve an elliptical orbit. This technique can be used to change the shape of an orbit for various purposes, such as increasing the range of a satellite or altering the trajectory of a spacecraft.

3. Can doubling speed in circular orbit lead to a stable elliptical orbit?

Yes, doubling speed in circular orbit can lead to a stable elliptical orbit. However, the stability of the orbit will also depend on other factors such as the mass and gravitational pull of the objects involved.

4. How does the distance between the object and the center of the orbit change when the speed is doubled?

When the speed is doubled in a circular orbit, the distance between the object and the center of the orbit will increase when the object moves away from the center, and decrease when the object moves closer to the center.

5. What are some potential applications of doubling speed in circular orbit to attain elliptical orbit?

Doubling speed in circular orbit to attain elliptical orbit has various applications in space exploration and satellite technology. For example, it can be used to launch satellites into higher orbits, change the trajectory of spacecraft, and increase the range of communication and observation satellites.

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