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**1. 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.

**2. Homework Equations**

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

##\frac{-GMm}{2a} = \frac{1}{2}mv^2 - \frac{GMm}{r}##

**3. 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?