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Homework Statement
a satellite is in a circular orbit a distance $h$ above the surface of the Earth with speed $v_0$, booster rockets are fired which double the speed of the satellite without changing the direction. Find the subsequent orbit.
Homework Equations
The Attempt at a Solution
Before the rocket boost, If we use the radial motion equation we can find the total energy of the system, that is given by:
$$E = \frac{1}{2}\dot{r}^2 + V + L^2/2r^2 = \frac{v_0^2}{2} - \frac{\gamma}{R_e + h}$$
Where I have assumed the presence of an attractive inverse square law.
and $L = rv_0$.
How can I continue from here? surely when the velocity increases, it isn't going to continue in a circular orbit.
After the boost, we have $$\frac{1}{2}\dot{r}^2 + V + L^2/2r^2 = \frac{1}{2}\dot{r}^2 - \frac{L^2}{2r^2} - \frac{\gamma}{r} = E $$we can write
$$\frac{1}{2}4v_0^2 - \frac{L^2}{2r^2} - \frac{\gamma}{r} = E $$
then equating
I obtain
$$\frac{L^2}{2r^2} + \frac{\gamma}{r} = \frac{\gamma}{R_e + h} + \frac{3}{2v_0^2}$$
I don't see how the solutions to the above equation describe the orbit