# Doubling speed in circular orbit to attain elliptical orbit

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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?

Mentor
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?

ln(
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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Mentor
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.