- #1

Zatman

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## Homework Statement

It is required to put a satellite of mass m into orbit with apogee 2.5 times the radius of the planet of mass M. The satellite is to be launched from the surface with speed v

_{0}at an angle of 30° to the local vertical.

Use conservation of energy and angular momentum to show that

[itex]v_0^2 = \frac{5GM}{4R}[/itex]

assuming that the planet is spherical, not rotating and atmospheric effects can be ignored.

## Homework Equations

[itex]L = mr^2\dot{\theta}[/itex]

[itex]V = -\frac{GMm}{r}[/itex]

where r is the distance from the centre of the planet to the satellite's position at time t.

## The Attempt at a Solution

At t = 0, the angular momentum is L = mRv

_{0}sin(30) = (1/2)mRv

_{0}. Therefore:

[itex]\dot{\theta} = \frac{L}{mr^2} = \frac{Rv_0}{2r^2}[/itex]

Total energy at time t is:

[itex]E = \frac{1}{2}mv^2 - \frac{GMm}{r}[/itex]

[itex]E = \frac{1}{2}mr^2\dot{\theta}^2 - \frac{GMm}{r}[/itex]

[itex]E = \frac{mR^2v_0^2}{8r^2} -\frac{GMm}{r}[/itex]

I don't really know what to do with these equations. At the apogee, dr/dt = 0, so I could differentiate the E equation, but then I just get 0 = 0. Also I notice that E should be the same for all r as well as t, but clearly E(r) is not constant, so is the energy equation wrong?

Any help would be greatly appreciated.