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.
