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The applet should be helpful in appreciating the Kepler's laws.
Let me make the initial velocity parameter clear. It is being expressed in terms of the velocity for circular orbit Vo. You should be able to get a circular orbit by choosing Vo as the velocity of projection and 90 degrees for angle of projection. This would be
This value is about 8 km/s for close earth orbits. ( Use R, the radius of the earth as the distance and an angle of projection of 90 degrees to see this). Note that The magnitude of velocity during subsequent motion changes if it is not a circular orbit.
You should be able to see the independence of time period on eccentricity by changing the angle of projection (Letting other parameters remain the same). It should be remembered that the major axis depends only on the energy and time period only the on the major axis. Since you are choosing a certain value for the distance from the earth for projection and are projecting at the same velocity at different angles the energy is same and hence the major axis. I am putting down the relevant relations here for your reference
and this in our case is equal to
The energy at the intial moment is dependent on r and v only and is unchanged by changing the angle of projection. The time period for a given orbit is
Try experimenting with cirucular orbits of radii in the ratio 1:4. You should be able to relate to the result of the relation above.
You could also see interesting situations of angular momentum conservation. Try choosing an angle of projection of 60 degrees when radius vector is 2*R and the initial velocity Vo. You would see the satellite grazing past the surface of the earth in its path. (Can you see why?) You should be able to see other possibilities.