- #1

- 94

- 0

## 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